A reduction of the spectrum problem for odd sun systems and the prime case

TL;DR

This paper extends the existence results for $k$-sun decompositions of complete graphs with odd $k$, showing that if such decompositions exist within a specific range, they exist for all larger admissible values.

Contribution

It adapts a method from odd cycle systems to reduce the spectrum problem for odd sun systems, establishing a broad existence criterion.

Findings

Existence of $k$-sun systems for all admissible $v \, \geq 6k$ under certain conditions.

Reduction of the spectrum problem to a finite range for odd sun systems.

Extension of known results from small cases to all larger admissible cases.

Abstract

A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. The existence problem for $k$-sun decompositions of $K_v$, with $k$ odd, has been solved only when $k=3$ or $5$. By adapting a method used by Hoffmann, Lindner and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a $k$-sun system of $K_v$ ($k$ odd) whenever $v$ lies in the range $2k< v < 6k$ and satisfies the obvious necessary conditions, then such a system exists for every admissible $v\geq 6k$.

Tables

Table 1. Table 1. From S i subscript 𝑆 𝑖 S_{i} to T i subscript 𝑇 𝑖 T_{i} .

$𝒊$	${𝑽}_{𝒊}\to {𝑽}_{𝒊}^{\prime }$	${𝑴}_{𝒊}$	${𝑵}_{𝒊}$	$𝝂$
$1$	$(x_2,y_3)\to ({\infty }_1^{\prime },{\infty }_2^{\prime })$	$\{x_1,x_2\},\{\overline{x_2},y_3\}$ $\{y_3,y_4\},\{y_3,\overline{y_4}\}$	$\{{\infty }_1^{\prime },x_1\},\{{\infty }_2^{\prime },\overline{x_2}\}$ $\{{\infty }_2^{\prime },y_4\},\{{\infty }_2^{\prime },\overline{y_4}\}$	2,3
$2$	$(\overline{x_2},y_3)\to ({\infty }_1^{\prime },{\infty }_2^{\prime })$	$\{\overline{x_1},\overline{x_2}\},\{x_2,y_3\}$	$\{{\infty }_1^{\prime },\overline{x_1}\},\{{\infty }_2^{\prime },x_2\}$	2
$2$	$(\overline{x_2},y_3,\overline{y_3})\to ({\infty }_1^{\prime },{\infty }_2^{\prime },{\infty }_3^{\prime })$	$\{\overline{x_1},\overline{x_2}\},\{x_2,y_3\}$ $\{x_2,\overline{y_3}\},\{\overline{y_3},\overline{y_4}\}$ $\{\overline{y_3},y_4\}$	$\{{\infty }_1^{\prime },\overline{x_1}\},\{{\infty }_2^{\prime },x_2\}$ $\{{\infty }_3^{\prime },x_2\},\{{\infty }_3^{\prime },\overline{y_4}\}$ $\{{\infty }_3^{\prime },y_4\}$	3
$3$	$y_2\to {\infty }_1^{\prime }$	$\{y_1,y_2\}$	$\{{\infty }_1^{\prime },y_1\}$	2,3
$4$	$\overline{y_2}\to {\infty }_1^{\prime }$	$\{\overline{y_1},\overline{y_2}\}$	$\{{\infty }_1^{\prime },\overline{y_1}\}$	2,3
$3+2j$	$\varnothing$	$\varnothing$	$\varnothing$	2,3
$4+2j$	$\varnothing$	$\varnothing$	$\varnothing$	2
$4+2j$	$\overline{c_{j+1}}\to {\infty }_3^{\prime }$	$\{\overline{c_j},\overline{c_{j+1}}\}$	$\{{\infty }_3^{\prime },\overline{c_j}\}$	3

Equations

$$\Gamma +w:=\Gamma \cup K_{V(\Gamma ),W},$$

$$\left(\begin{array}[]{ccccc}x_{1}&x_{2}&\ldots&x_{k-1}&x_{k}\\ x^{\prime}_{1}&x^{\prime}_{2}&\ldots&x^{\prime}_{k-1}&x^{\prime}_{k}\end{array}\right),$$

$$\ast$$

$${\Delta }_{ij}\Gamma$$

$$\displaystyle\cup\ \big{\{}a_{h}-a_{h+1}\mid x_{h}=(a_{h},j),x_{h+1}=(a_{h+1},i),1\leq{h}\leq{k/s}\big{\}};$$

$${\Delta }_{ij}\Gamma$$

$$\displaystyle\cup\ \big{\{}a_{h}-a^{\prime}_{h}\mid x_{h}=(a_{h},j),x^{\prime}_{h}=(a^{\prime}_{h},i),1\leq{h}\leq{k/s}\big{\}}.$$

$$\big{\langle}D_{ij}\mid 0\leq i\leq j\leq m-1\big{\rangle}$$

$$a_i$$

$$b_j$$

$$P$$

$$Q$$

$$\begin{array}[]{lll}\Delta_{00}P=\pm[1+\epsilon,s+\epsilon],&\Delta_{01}P=\{0\},&\Delta_{11}P=\pm[1+\epsilon,s^{\prime}+\epsilon],\\ \Delta_{00}Q=\varnothing,&\Delta_{01}Q=[1,t-1],&\Delta_{11}Q=\varnothing,\end{array}$$

$$\pm (a_{s-u}-a_{s-u-1})=\pm (a_{s+u+2}-a_{s+u+1})=\pm (u+ϵ+1,0).$$

$$\text{Dev({c1,c2})=Γ and Orb(C) is a k-cycle system of Γ+ℓ}.$$

$$\text{Dev({c1,c2}) is the k-cycle of Γ and Orb(C) is a k-cycle system of Γ+ℓ}.$$

$$c_{h,1}=\big{(}0,1-h\big{)},\;c_{h,2}=\big{(}h\ell,0\big{)},\;\text{and}\;c_{j}=\begin{cases}\left(\frac{j-1}{2},1\right)&\text{if $j\in[3,\ell+1]$ is odd},\\ \left(\frac{j}{2},0\right)&\text{if $j\in[4,\ell+1]$ is even}.\end{cases}$$

$$\overline{x}=\{\begin{array}{ll}(a,2-j)& \text{if x=(a,j)∈ZM×{0,2},}\\ (a,4-j)& \text{if x=(a,j)∈ZM×{1,3},}\\ {\infty }_{h+u}& \text{if x=∞h}.\end{array}$$

$$E(\Gamma[2])=\big{\{}\{x,y\},\{x,\overline{y}\},\{\overline{x},y\},\{\overline{x},\overline{y}\}\mid\{x,y\}\in E(\Gamma)\big{\}},$$

$$I=\big{\{}\{x,\overline{x}\}\mid x\in\mathbb{Z}_{M}\times\{0,1\}\big{\}}.$$

$$S=\left(\begin{array}[]{cccc}s_{1}&\ldots&s_{k-1}&s_{k}\\ \overline{s_{2}}&\ldots&\overline{s_{k}}&\overline{s_{1}}\end{array}\right)$$

$$\sigma(C)=\left(\begin{array}[]{cccc}c_{1}&\ldots&c_{k-1}&c_{k}\\ \overline{c_{2}}&\ldots&\overline{c_{k}}&\overline{c_{1}}\end{array}\right).$$

$$K_{4k}+(2w_1+w_2)$$

$$=({\Gamma }_1[2]+2w_1)\oplus ({\Gamma }_2^{\ast }[2]+w_2)=({\Gamma }_1+w_1)[2]\oplus ({\Gamma }_2^{\ast }[2]+w_2).$$

$$\displaystyle\Gamma_{1}=\big{\langle}[2,\ell],[k-2r-2,k-1],[2,\ell-1]\big{\rangle}\;\;\mbox{and}\;\;\Gamma_{2}=\big{\langle}\{1\},[0,k-2r-3],\{1,\ell\}\big{\rangle}.$$

$$\big{\langle}[2,\ell-1],\varnothing,\varnothing\big{\rangle},\hskip 8.5359pt\big{\langle}\varnothing,\varnothing,[2,\ell-1]\big{\rangle},\hskip 8.5359pt\big{\langle}\{\ell\},[k-2r-2,k-1],\varnothing\big{\rangle},$$

$$C_1$$

$$\Omega_{2}^{*}[2]+w_{2}=\big{(}\Gamma_{2}+(\ell+r)\big{)}[2]\oplus(\Gamma_{3}+\ell)[2]\oplus(I+\nu)$$

$$Orb(A)\cup Orb(B)$$

$$Dev(\{x_1,x_2\})$$

$$Dev(\{y_1,y_2\})$$

$$\begin{array}{rl}& \text{Γ3+ℓ has a k-cycle system {F1,F2,…,Fk} such that}\\ & \text{Fj=(cj,cj+1,fj,3,fj,4,…,fj,k) for every j∈[1,k] (with ck+1=c1).}\end{array}$$

$$S_1=\sigma (x_1,\overline{x_2},y_3,y_4,a_5,\dots ,a_k),S_3=\sigma (y_1,\overline{y_2},b_3,\dots ,b_k),$$

$$S_{3+2j}=\sigma (c_j,\overline{c_{j+1}},f_{j,3},f_{j,4},\dots ,f_{j,k})\text{for j∈[1,k]},\text{and}$$

Full text

By adapting a method used by Hoffmann, Lindner and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a $k$-sun system of $K_{v}$ ($k$ odd) whenever $v$ lies in the range $2k<v<6k$ and satisfies the obvious necessary conditions, then such a system exists for every admissible $v\geq 6k$.

Furthermore, we give a complete solution whenever $k$ is an odd prime.

1. Introduction

We denote by $V(\Gamma)$ and $E(\Gamma)$ the set of vertices and the list of edges of a graph $\Gamma$, respectively. Also, we denote by $\Gamma+w$ the graph obtained by adding to $\Gamma$ an independent set $W=\{\infty_{i}\mid 1\leq{i}\leq{w}\}$ of $w\geq 0$ vertices each adjacent to every vertex of $\Gamma$, namely,

[TABLE]

where $K_{V(\Gamma),W}$ is the complete bipartite graph with parts $V(\Gamma)$ and $W$. Denoting by $K_{v}$ the complete graph of order $v$, it is clear that $K_{v}+1$ is isomorphic to $K_{v+1}$.

We denote by $x_{1}\sim x_{2}\sim\ldots\sim x_{k}$ the path with edges $\{x_{i-1},x_{i}\}$ for $2\leq{i}\leq{k}$. By adding the edge $\{x_{1},x_{k}\}$ when $k\geq 3$, we obtain a cycle of length $k$ (briefly, a $k$-cycle) denoted by $(x_{1},x_{2},\ldots,x_{k})$. A $k$-cycle with further $v-k\geq 0$ isolated vertices will be referred to as a $k$-cycle of order $v$. By adding to $(x_{1},x_{2},\ldots,x_{k})$ an independent set of edges \big{\{}\{x_{i},x^{\prime}_{i}\}\mid 1\leq{i}\leq{k}\big{\}}, we obtain the $k$-sun on $2k$ vertices (sometimes referred to as $k$-crown graph) denoted by

[TABLE]

whose edge-set is therefore \big{\{}\{x_{i},x_{i+1}\},\{x_{i},x^{\prime}_{i}\}\mid 1\leq{i}\leq{k}\big{\}}, where $x_{k+1}=x_{1}$.

A decomposition of a graph $K$ is a set $\{\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{t}\}$ of subgraphs of $K$ whose edge-sets between them partition the edge-set of $K$; in this case, we briefly write $K=\oplus_{i=1}^{t}\Gamma_{i}$. If each $\Gamma_{i}$ is isomorphic to $\Gamma$, we speak of a $\Gamma$-decomposition of $K$. If $\Gamma$ is a $k$-cycle (resp., $k$-sun), we also speak of a $k$-cycle system (resp., $k$-sun system) of $K$.

In this paper we study the existence problem for $k$-sun systems of $K_{v}$ ($v>1$). Clearly, for such a system to exist we must have

[TABLE]

As far as we know, this problem has been completely settled only when $k=3,5$ [8, 10], $k=4,6,8$ [12], and when $k=10,14$ or $2^{t}\geq 4$ [9]. It is important to notice that, as a consequence of a general result proved in [14], condition ($\ast$ ‣ 1) is sufficient whenever $v$ is large enough with respect to $k$. These results seem to suggest the following.

Conjecture 1**.**

Let $k\geq 3$ and $v>1$. There exists a $k$-sun system of $K_{v}$ if and only if ($\ast$ ‣ 1) holds.

Our constructions rely on the existence of $k$-cycle systems of $K_{v}$, a problem that has been completely settled in [1, 4, 5, 11, 13]. More precisely, [4] and [11] reduce the problem to the orders $v$ in the range $k\leq v<3k$, with $v$ odd. These cases are then solved in [1, 13]. For odd $k$, an alternative proof based on $1$-rotational constructions is given in [5]. Further results on $k$-cycle systems of $K_{v}$ with an automorphism group acting sharply transitively on all but at most one vertex can be found in [2, 6, 7, 15].

The main results of this paper focus on the case where $k$ is odd. By adapting a method used in [11] to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a $k$-sun system of $K_{v}$ ($k$ odd) whenever $v$ lies in the range $2k<v<6k$ and satisfies the obvious necessary conditions, then such a system exists for every admissible $v\geq 6k$. In other words, we show the following.

Theorem 1.1**.**

Let $k\geq 3$ be an odd integer and $v>1$. Conjecture 1 is true if and only if there exists a $k$-sun system of $K_{v}$ for all $v$ satisfying the necessary conditions in ($\ast$ ‣ 1) with $2k<v<6k$.

We would like to point out that we strongly believe the reduction methods used in [4, 11] could be further developed to reduce the spectrum problem of other types of graph decompositions of $K_{v}$.

In Section 6, we construct $k$-sun systems of $K_{v}$ for every odd prime $k$ whenever $2k<v<6k$ and ($\ast$ ‣ 1) holds. Therefore, as a consequence of Theorem 1.1, we solve the existence problem for $k$-sun systems of $K_{v}$ whenever $k$ is an odd prime.

Theorem 1.2**.**

For every odd prime $p$ there exists a $p$-sun system of $K_{v}$ with $v>1$ if and only if $v\geq 2p$ and $v(v-1)\equiv 0\pmod{4p}$.

Both results rely on the difference methods described in Section 2. These methods are used in Section 3 to construct specific $k$-cycle decompositions of some subgraphs of $K_{2k}+w$, which we then use in Section 4 to build $k$-sun systems of $K_{4k}+n$. This is the last ingredient we need in Section 5 to prove Theorem 1.1. Difference methods are finally used in Section 6 to construct $k$-sun systems of $K_{v}$ for every odd prime $k$ whenever $2k<v<6k$ and ($\ast$ ‣ 1) holds.

2. Preliminaries

Henceforward, $k\geq 3$ is an odd integer, and $\ell=\frac{k-1}{2}$. Also, given two integers $a\leq b$, we denote by $[a,b]$ the interval containing the integers $\{a,a+1,\ldots,b\}$. If $a>b$, then $[a,b]$ is empty.

In our constructions we make extensive use of the method of partial mixed differences which we now recall but limited to the scope of this paper.

Let $G$ be an abelian group of odd order $n$ in additive notation, let $W=\{\infty_{u}\mid 1\leq{u}\leq{w}\}$, and denote by $\Gamma$ a graph with vertices in $V=(G\times[0,m-1])\ \cup\ W$. For any permutation $f$ of $V$, we denote by $f(\Gamma)$ the graph obtained by replacing each vertex of $\Gamma$, say $x$, with $f(x)$. Letting $\tau_{g}$, with $g\in G$, be the permutation of $V$ fixing each $\infty_{u}\in W$ and mapping $(x,i)\in G\times[0,m-1]$ to $(x+g,i)$, we call $\tau_{g}$ the translation by $g$ and $\tau_{g}(\Gamma)$ the related translate of $\Gamma$.

We denote by $Orb_{G}(\Gamma)=\{\tau_{g}(\Gamma)\mid g\in G\}$ the $G$-orbit of $\Gamma$, that is, the set of all distinct translates of $\Gamma$, and by $Dev_{G}(\Gamma)=\bigcup_{g\in G}\tau_{g}(\Gamma)$ the graph union of all translates of $\Gamma$. Further, by $Stab_{G}(\Gamma)=\{g\in G\mid\tau_{g}(\Gamma)=\Gamma\}$ we denote the $G$-stabilizer of $\Gamma$, namely, the set of translations fixing $\Gamma$. We recall that $Stab_{G}(\Gamma)$ is a subgroup of $G$, hence $s=|Stab_{G}(\Gamma)|$ is a divisor of $n=|G|$. Henceforward, when $G=\mathbb{Z}_{k}$, we will simply write $Orb(\Gamma)$, $Dev(\Gamma)$, and $Stab(\Gamma)$.

Suppose now that $\Gamma$ is either a $k$-cycle or a $k$-sun with vertices in $V$. For every $i,j\in[0,m-1]$, the list of $(i,j)$-differences of $\Gamma$ is the multiset $\Delta_{ij}\Gamma$ defined as follows:

1. (1)

if $\Gamma=(x_{1},x_{2},\ldots,x_{k})$, then

[TABLE] 2. (2)

if \Gamma=\left(\begin{array}[]{cccc}x_{1}&x_{2}&\ldots&x_{k}\\ x^{\prime}_{1}&x^{\prime}_{2}&\ldots&x^{\prime}_{k}\end{array}\right), then

[TABLE]

We notice that when $s=1$ we find the classic concept of list of differences. Usually, one speaks of pure or mixed differences according to whether $i=j$ or not, and when $m=1$ we simply write $\Delta\Gamma$. This concept naturally extends to a family $\mathcal{F}$ of graphs with vertices in $V$ by setting $\Delta_{ij}\mathcal{F}=\bigcup_{\Gamma\in{\mathcal{F}}}\Delta_{ij}\Gamma$. Clearly, $\Delta_{ij}\Gamma=-\Delta_{ji}\Gamma$, hence $\Delta_{ij}{\mathcal{F}}=-\Delta_{ji}{\mathcal{F}}$, for every $i,j\in[0,m-1]$.

We also need to define the list of neighbours of $\infty_{u}$ in ${\mathcal{F}}$, that is, the multiset $N_{{\mathcal{F}}}(\infty_{u})$ of the vertices in $V$ adjacent to $\infty_{u}$ in some graph $\Gamma\in{\mathcal{F}}$.

Finally, we introduce a special class of subgraphs of $K_{mn}$. To this purpose, we take $V(K_{mn})=G\times[0,m-1]$. Letting $D_{ii}\subseteq G\setminus\{0\}$ for every $0\leq i\leq m-1$, and $D_{ij}\subseteq G$ for every $0\leq i<j\leq m-1$, we denote by

[TABLE]

the spanning subgraph of $K_{mn}$ containing exactly the edges \big{\{}(g,i),(g+d,j)\big{\}} for every $g\in G$, $d\in D_{ij}$, and $0\leq i\leq j\leq m-1$. The reader can easily check that this graph remains unchanged if we replace any set $D_{ii}$ with $\pm D_{ii}$.

The following result, standard in the context of difference families, provides us with a method to construct $\Gamma$-decompositions for subgraphs of $K_{mn}+w$.

Proposition 2.1**.**

Let $G$ be an abelian group of odd order $n$, let $m$ and $w$ be non-negative integers, and denote by $\mathcal{F}$ a family of $k$-cycles (resp., $k$-suns) with vertices in $(G\times[0,m-1])\ \cup\ \{\infty_{u}\mid u\in\mathbb{Z}_{w}\}$ satisfying the following conditions:

1. (1)

$\Delta_{ij}\mathcal{F}$* has no repeated elements, for every $0\leq i\leq j<m$;* 2. (2)

N_{\mathcal{F}}(\infty_{u})=\big{\{}(g_{u,i},i)\mid 0\leq i<m,g_{u,i}\in G\big{\}}* for every $1\leq{u}\leq{w}$.*

Then $\bigcup_{\Gamma\in\mathcal{F}}Orb_{G}(\Gamma)=\{\tau_{g}(\Gamma)\mid g\in G,\Gamma\in{\mathcal{F}}\}$ is a $k$-cycle (resp., $k$-sun) system of $\langle\Delta_{ij}{\mathcal{F}}\mid 0\leq{i}\leq{j}\leq m-1\rangle+w$.

Proof.

Let ${\mathcal{F}}^{*}=\bigcup_{\Gamma\in\mathcal{F}}Orb_{G}(\Gamma)$, $K=\langle\Delta_{ij}{\mathcal{F}}\mid 0\leq{i}\leq{j}\leq m-1\rangle$, and let $\epsilon$ be an edge of $K+w$. We are going to show that $\epsilon$ belongs to exactly one graph of ${\mathcal{F}}^{*}$.

If $\epsilon\in E(K)$, by recalling the definition of $K$ we have that $\epsilon=\{(g,i),(g+d,j)\}$ for some $g\in G$ and $d\in\Delta_{ij}{\mathcal{F}}$, with $0\leq i\leq j<m$. Hence, there is a graph $\Gamma\in{\mathcal{F}}$ such that $d\in\Delta_{ij}\Gamma$. This means that $\Gamma$ contains the edge $\epsilon^{\prime}=\{(g^{\prime},i),(g^{\prime}+d,j)\}$ for some $g^{\prime}\in G$, therefore $\epsilon=\tau_{g-g^{\prime}}(\epsilon^{\prime})\in\tau_{g-g^{\prime}}(\Gamma)\in{\mathcal{F}}^{*}$. To prove that $\epsilon$ only belongs to $\tau_{g-g^{\prime}}(\Gamma)$, let $\Gamma^{\prime}$ be any graph in ${\mathcal{F}}$ such that $\epsilon\in\tau_{x}(\Gamma^{\prime})$, for some $x\in G$. Since translations preserve differences, we have that $d\in\Delta_{ij}\tau_{x}(\Gamma^{\prime})=\Delta_{ij}\Gamma^{\prime}$. Considering that $d\in\Delta_{ij}\Gamma\ \cap\ \Delta_{ij}\Gamma^{\prime}$ and, by condition (1), $\Delta_{ij}{\mathcal{F}}$ has no repeated elements, we necessarily have that $\Gamma^{\prime}=\Gamma$, hence $\tau_{-x}(\epsilon)\in\Gamma$. Again, since $\Delta_{ij}\Gamma$ has no repeated elements (condition (1)), and considering that $\epsilon^{\prime}$ and $\tau_{-x}(\epsilon)$ are edges of $\Gamma$ that yield the same differences, then $\tau_{-x}(\epsilon)=\epsilon^{\prime}=\tau_{g^{\prime}-g}(\epsilon)$, that is, $\tau_{g^{\prime}-g+x}(\epsilon)=\epsilon$. Since $G$ has odd order, it has no element of order $2$, hence $g^{\prime}-g+x=0$, that is, $x=g-g^{\prime}$, therefore $\tau_{g-g^{\prime}}(\Gamma)$ is the only graph of ${\mathcal{F}}^{*}$ containing $\epsilon$.

Similarly, we show that every edge of $(K+w)\setminus{K}$ belongs to exactly one graph of ${\mathcal{F}}^{*}$. Let $\epsilon=\{\infty_{u},(g,i)\}$ for some $u\in\mathbb{Z}_{w}$ and $(g,i)\in G\times[0,m-1]$. By assumption, there is a graph $\Gamma\in{\mathcal{F}}^{*}$ containing the edge $\epsilon^{\prime}=\{\infty_{u},(g_{u,i},i)\}$ with $g_{u,i}\in G$. Hence, $\epsilon=\tau_{g-g_{u,i}}(\epsilon^{\prime})\in\tau_{g-g_{u,i}}(\Gamma)$. Finally, if $\epsilon\in\tau_{x}(\Gamma^{\prime})$ for some $x\in G$ and $\Gamma^{\prime}\in{\mathcal{F}}$, then $\{\infty_{u},(g-x,i)\}=\tau_{-x}(\epsilon)\in\Gamma^{\prime}$. Since condition (2) implies that $N_{\mathcal{F}}(\infty_{u})$ contains exactly one pair from $G\times\{i\}$, we necessarily have that $\Gamma=\Gamma^{\prime}$ and $x=g-g_{u,i}$; therefore, there is exactly one graph of ${\mathcal{F}}^{*}$ containing $\epsilon$. Condition (2) also implies that $N_{\mathcal{F}}(\infty_{u})$ is disjoint from $\{\infty_{u}\mid u\in\mathbb{Z}_{w}\}$, and this guarantees that no graph in $\mathcal{F}^{*}$ contains edges joining two infinities. Therefore, $\mathcal{F}^{*}$ is the desired decomposition of $K+w$. ∎

Considering that $K_{mn}=\langle D_{ij}\mid 0\leq i\leq j\leq m-1\rangle$ if and only if $\pm D_{ii}=G\setminus\{0\}$ for every $i\in[0,m-1]$, and $D_{ij}=G$ for every $0\leq i<j\leq m-1$, the proof of the following corollary to Proposition 2.1 is straightforward.

Corollary 2.2**.**

Let $G$ be an abelian group of odd order $n$, let $m$ and $w$ be non-negative integers, and denote by $\mathcal{F}$ a family of $k$-cycles (resp., $k$-suns) with vertices in $(G\times[0,m-1])\ \cup\ \{\infty_{u}\mid u\in\mathbb{Z}_{w}\}$ satisfying the following conditions:

1. (1)

\Delta_{ij}\mathcal{F}=\begin{cases}G\setminus\{0\}&\text{if 0\leq i=j\leq m-1};\\ G&\text{if 0\leq i<j\leq m-1};\end{cases}** 2. (2)

N_{\mathcal{F}}(\infty_{u})=\big{\{}(g_{u,i},i)\mid 0\leq i<m,g_{u,i}\in G\big{\}}* for every $1\leq{u}\leq{w}$.*

Then $\bigcup_{\Gamma\in\mathcal{F}}Orb_{G}(\Gamma)$ is a $k$-cycle (resp., $k$-sun) system of $K_{mn}+w$.

3. Constructing $k$-cycle systems of $\langle D_{00},D_{01},D_{11}\rangle+w$

In this section, we recall and generalize some results from [11] in order to provide conditions on $D_{00},D_{01},D_{11}\subseteq\mathbb{Z}_{k}$ that guarantee the existence of a $k$-cycle system for the subgraph $\langle D_{00},D_{01},D_{11}\rangle+w$ of $K_{2k}+w$, where $V(K_{2k})=\mathbb{Z}_{k}\times\{0,1\}$.

We recall that every connected 4-regular Cayley graph over an abelian group has a Hamilton cycle system [3] and show the following.

Lemma 3.1**.**

Let $[a,b],[c,d]\subseteq[1,\ell]$. The graph \big{\langle}\left[a,b\right],\varnothing,\left[c,d\right]\big{\rangle} has a $k$-cycle system whenever both $[a,b]$ and $[c,d]$ satisfy the following condition: the interval has even size or contains an integer coprime with $k$.

Proof.

The graph \big{\langle}\left[a,b\right],\varnothing,\left[c,d\right]\big{\rangle} decomposes into \big{\langle}\left[a,b\right],\varnothing,\varnothing\big{\rangle} and \big{\langle}\varnothing,\varnothing,\left[c,d\right]\big{\rangle}. The first one is the Cayley graph $\Gamma=\mathrm{Cay}(\mathbb{Z}_{k},[a,b])$ with further $k$ isolated vertices, while the second one is isomorphic to \big{\langle}\left[c,d\right],\varnothing,\varnothing\big{\rangle}. Therefore, it is enough to show that $\Gamma$ has a $k$-cycle system.

Note that $\Gamma$ decomposes into the subgraphs $\mathrm{Cay}(\mathbb{Z}_{k},D_{i})$, for $0\leq i\leq t$, whenever the sets $D_{i}$ between them partition $\left[a,b\right]$. By assumption, $[a,b]$ has even size or contains an integer coprime with $k$. Therefore, we can assume that for every $i>0$ the set $D_{i}$ is a pair of integers at distance 1 or 2, and $D_{0}$ is either empty or contains exactly one integer coprime with $k$. Clearly, $\mathrm{Cay}(\mathbb{Z}_{k},D_{0})$ is either the empty graph or a $k$-cycle, and the remaining $\mathrm{Cay}(\mathbb{Z}_{k},D_{i})$ are 4-regular Cayley graphs. Also, for every $i>0$ we have that $D_{i}$ is a generating set of $\mathbb{Z}_{k}$ (since $k$ is odd and $D_{i}$ contains integers at distance 1 or 2), hence the graph $\mathrm{Cay}(\mathbb{Z}_{k},D_{i})$ is connected. It follows that each $\mathrm{Cay}(\mathbb{Z}_{k},D_{i})$, with $i>0$, decomposes into two $k$-cycles, thus the assertion is proven. ∎

Lemma 3.2**.**

Let $S\subseteq\{2i-1\mid 1\leq i\leq\ell\}$. Then there exist $k$-cycle systems for the graphs \big{\langle}\{\ell\},S\ \cup\ (S+1),\varnothing\big{\rangle} and \big{\langle}\{\ell\},(S+1)\ \cup\ (S+2),\varnothing\big{\rangle}.

Proof.

We note that the result is trivial when $S=\varnothing$, since \big{\langle}\{\ell\},\varnothing,\varnothing\big{\rangle} is a $k$-cycle.

The existence of a $k$-cycle system of \Gamma=\big{\langle}\{\ell\},S\ \cup\ (S+1),\varnothing\big{\rangle} has been proven in [11, Lemma 3] when $S\subseteq\{2i-1\mid 1\leq i\leq\ell\}$. Consider now the permutation $f$ of $\mathbb{Z}_{k}\times\{0,1\}$ fixing $\mathbb{Z}_{k}\times\{0\}$ pointwise, and mapping $(i,1)$ to $(i+1,1)$ for every $i\in\mathbb{Z}_{k}$. It is not difficult to check that f(\Gamma)=\big{\langle}\{\ell\},(S+1)\ \cup\ (S+2),\varnothing\big{\rangle} which is therefore isomorphic to $\Gamma$, and hence it has a $k$-cycle system. ∎

Lemma 3.3**.**

Let $r,s$ and $s^{\prime}$ be integers such that $1\leq s\leq s^{\prime}\leq\min\{s+1,\ell\}$, and $0<r\not\equiv s+s^{\prime}\pmod{2}$. Also, let $D\subseteq[0,k-1]$ be a non-empty interval of size $k-(s+s^{\prime}+2r)$. Then there is a cycle $C=(x_{1},x_{2},\ldots,x_{k})$ of \Gamma=\big{\langle}[1+\epsilon,s+\epsilon],D,[1+\epsilon,s^{\prime}+\epsilon]\big{\rangle}+r, for every $\epsilon\in\{0,1\}$, such that $Orb(C)$ is a $k$-cycle system of $\Gamma$. Furthermore, if $u=0$ or $u=1-\epsilon=1\leq s-1$, then

1. (1)

Dev\big{(}\{x_{2-u},x_{3-u}\}\big{)}* is a $k$-cycle with vertices in $\mathbb{Z}_{k}\times\{0\}$;* 2. (2)

Dev\big{(}\{x_{4+u},x_{5+u}\}\big{)}* is a $k$-cycle with vertices in $\mathbb{Z}_{k}\times\{1\}$.*

Proof.

Set $t=k-(s+s^{\prime}+2r)$ and let \Omega=\big{\langle}[1+\epsilon,s+\epsilon],[0,t-1],[1+\epsilon,s^{\prime}+\epsilon]\big{\rangle}+r. For $i\in[0,s+s^{\prime}+1]$ and $j\in[0,t+r-1]$, let $a_{i}$ and $b_{j}$ be the elements of $\mathbb{Z}_{k}\times\{0,1\}$ defined as follows:

[TABLE]

Since the elements $a_{i}$ and $b_{j}$ are pairwise distinct, except for $a_{0}=b_{0}$ and $a_{s+s^{\prime}+1}=b_{t+r-1}$, then the union $F$ of the following two paths is a $k$-cycle:

[TABLE]

Since $\Delta_{ij}F=\Delta_{ij}P\ \cup\ \Delta_{ij}Q$, for $i,j\in\{0,1\}$, where

[TABLE]

and considering that $N_{F}(\infty_{h})=N_{Q}(\infty_{h})=\{b_{t+h-2},b_{t+h-1}\}$ for every $h\in[1,r]$, Proposition 2.1 guarantees that $Orb(F)$ is a $k$-cycle system of $\Omega$. Furthermore, if $u=0$ or $u=1-\epsilon=1\leq s-1$, then

[TABLE]

Since $k$ is odd, we have that $Dev(\{a_{s-u-1},a_{s-u}\})$ and $Dev(\{a_{s+u+2},a_{s+u+1}\})$ are $k$-cycles with vertices in $\mathbb{Z}_{k}\times\{0\}$ and $\mathbb{Z}_{k}\times\{1\}$, respectively.

If $D=[g,g+t-1]$ is any interval of $[0,k-1]$ of size $t$, and $f$ is the permutation of $\mathbb{Z}_{k}\times\{0,1\}$ fixing $\mathbb{Z}_{k}\times\{0\}$ pointwise, and mapping $(i,1)$ to $(i+g,1)$ for every $i\in\mathbb{Z}_{k}$, one can check that $C=f(F)$ is the desired $k$-cycle of $\Gamma=f(\Omega)$. ∎

Lemma 3.4**.**

1. (1)

Let $\ell$ be odd. If $\Gamma$ is a $1$-factor of $K_{2k}$, then $\Gamma+\ell$ decomposes into $k$ cycles of length $k$, each of which contains exactly one edge of $\Gamma$. Furthermore, if \Gamma=\big{\langle}\varnothing,\{d\},\varnothing\big{\rangle}, then there exists a $k$-cycle $C=(c_{1},c_{2},\ldots,c_{k})$ of $\Gamma+\ell$, with $c_{1}\in\mathbb{Z}_{k}\times\{0\}$ and $c_{2}\in\mathbb{Z}_{k}\times\{1\}$, such that

[TABLE] 2. (2)

Let $\ell$ be even. If $\Gamma$ is a $k$-cycle of order $2k$, then $\Gamma+\ell$ decomposes into $k$ cycles of length $k$, each of which contains exactly one edge of $\Gamma$. Furthermore, if \Gamma=\big{\langle}\{d\},\varnothing,\varnothing\big{\rangle} and $d$ is coprime with $k$, then there exists a $k$-cycle $C=(c_{1},c_{2},\ldots,c_{k})$ of $\Gamma+\ell$, with $c_{1},c_{2}\in\mathbb{Z}_{k}\times\{0\}$, such that

[TABLE]

Proof.

Permuting the vertices of $K_{2k}$ if necessary, we can assume that $\Gamma$ is the $1$-factor \Gamma_{0}=\big{\langle}\varnothing,\{0\},\varnothing\big{\rangle} when $\ell$ is odd, and the $k$-cycle \Gamma_{1}=\big{\langle}\{\ell\},\varnothing,\varnothing\big{\rangle} (of order $2k$) when $\ell$ is even. For $h\in\{0,1\}$, let $C_{h}=(c_{h,1},c_{h,2},\infty_{1},c_{3},\infty_{2},c_{4},\ldots,\infty_{\ell-1},c_{\ell+1},\infty_{\ell})$ be the $k$-cycle of $\Gamma_{h}+\ell$, where

[TABLE]

Note that the sets $\Delta_{ij}C_{h}$ are empty, except for $\Delta_{01}C_{0}=\{0\}$ and $\Delta_{00}C_{1}=\{\pm\ell\}$. Also, the two neighbours of $\infty_{u}$ in $C_{h}$ belong to $\mathbb{Z}_{k}\times\{0\}$ and $\mathbb{Z}_{k}\times\{1\}$, respectively. Hence, Proposition 2.1 guarantees that $Orb(C_{h})$ is a $k$-cycle system of $\Gamma_{h}+\ell$, for $h\in\{0,1\}$. We finally notice that $Dev(\{c_{h,1},c_{h,2}\})=\Gamma_{h}$ (up to isolated vertices) and this completes the proof. ∎

The following result has been proven in [11].

Lemma 3.5**.**

Let $D\subseteq\left[1,\ell\right]$. The subgraph $\langle D,\{0\},D\rangle$ of $K_{2k}$ has a $1$-factorization.

Remark 3.6**.**

Considering the permutation $f$ of $\mathbb{Z}_{k}\times\{0,1\}$ such that $f(i,j)=(i,1-j)$, and a graph \Gamma=\big{\langle}D_{0},D_{1},D_{2}\big{\rangle}, we have that f(\Gamma)=\big{\langle}D_{2},-D_{1},D_{0}\big{\rangle}. Therefore, Lemmas 3.1 – 3.5 continue to hold when we replace $\Gamma$ by $f(\Gamma)$.

4. $k$-sun systems of $K_{4k}+n$

In this section we provide sufficient conditions for a $k$-sun system of $K_{4k}+n$ to exist, when $n\equiv 0,1\pmod{4}$. More precisely, we show the following.

Theorem 4.1**.**

Let $k\geq 7$ be an odd integer and let $n\equiv 0,1\pmod{4}$ with $2k<n<10k$, then there exists a $k$-sun system of $K_{4k}+n$, except possibly when

• •

$k=7$* and $n=20,21,32,33,44,45,56,57,64,65,68,69$,*

• •

$k=11$* and $n=100,101,112,113$.*

To prove Theorem 4.1, we start by introducing some notions and prove some preliminary results. Let $M$ be a positive integer and take $V(K_{2^{i}M})=\mathbb{Z}_{M}\times[0,2^{i}-1]$ and V(K_{2^{i}M}+w)=V(K_{2^{i}M})\ \cup\ \big{\{}\infty_{h}\mid h\in\mathbb{Z}_{w}\big{\}}, for $i\in\{1,2\}$ and $w>0$.

Now assume that $w=2u$, and let $x\mapsto\overline{x}$ be the permutation of $V(K_{4M}+2u)$ defined as follows:

[TABLE]

For any subgraph $\Gamma$ of $K_{4M}+2u$, we denote by $\overline{\Gamma}$ the graph (isomorphic to $\Gamma$) obtained by replacing each vertex $x$ of $\Gamma$ with $\overline{x}$.

Given a subgraph $\Gamma$ of $K_{2M}+u$, we denote by $\Gamma[2]$ the spanning subgraph of $K_{4M}+2u$ whose edge set is

[TABLE]

and let $\Gamma^{*}[2]=\Gamma[2]\oplus I$ be the graph obtained by adding to $\Gamma[2]$ the $1$-factor

[TABLE]

Note that, up to isolated vertices, $\Gamma[2]$ is the lexicographic product of $\Gamma$ with the empty graph on two vertices.

The proof of the following elementary lemma is left to the reader.

Lemma 4.2**.**

Let $\Gamma=\oplus_{i=1}^{n}\Gamma_{i}$ and let $w=\sum_{i=1}^{n}w_{i}$ with $w_{i}\geq 0$. If $\Gamma$ and the $\Gamma_{i}$s have the same vertex set (possibly with isolated vertices), then

1. (1)

$\Gamma+w=\oplus_{i=1}^{n}(\Gamma_{i}+w_{i})$; 2. (2)

$\Gamma[2]=\oplus_{i=1}^{n}\Gamma_{i}[2]$; 3. (3)

$(\Gamma+w)[2]=\Gamma[2]+2w$.

We start showing that if $C$ is a $k$-cycle, then $C[2]$ decomposes into two $k$-suns.

Lemma 4.3**.**

Let $C=(c_{1},c_{2},\ldots,c_{k})$ be a cycle with vertices in \big{(}\mathbb{Z}_{M}\times\{0,1\}\big{)}\ \cup\ \{\infty_{h}\mid h\in\mathbb{Z}_{u}\} and let $S$ be the $k$-sun defined as follows:

[TABLE]

where $s_{i}\in\{c_{i},\overline{c_{i}}\}$ for every $i\in[1,k]$. Then $C[2]=S\oplus\overline{S}$.

Proof.

It is enough to notice that $S$ contains the edges $\{s_{i},s_{i+1}\}$ and $\{s_{i},\overline{s_{i+1}}\}$, while $\overline{S}$ contains $\{\overline{s_{i}},\overline{s_{i+1}}\}$ and $\{\overline{s_{i}},s_{i+1}\}$, for every $i\in[1,k]$, where $s_{k+1}=s_{1}$ and $\overline{s_{k+1}}=\overline{s_{1}}$. ∎

Example 4.4**.**

In Figure 1 we have the graph $C_{7}[2]$ which can be decomposed into two $7$-suns $S$ and $\overline{S}$. The non-dashed edges are those of $S$, while the dashed edges are those of $\overline{S}$.

For every cycle $C=(c_{1},c_{2},\ldots,c_{k})$ with vertices in $\mathbb{Z}_{M}\times\{0,1\}$, we set

[TABLE]

Clearly, $C[2]=\sigma(C)\oplus\overline{\sigma(C)}$ by Lemma 4.3.

Lemma 4.5**.**

If $\mathcal{C}=\{C_{1},C_{2},\ldots,C_{t}\}$ is a $k$-cycle system of $\Gamma+u$, where $\Gamma$ is a subgraph of $K_{2M}$, and $S_{i}$ is a $k$-sun obtained from $C_{i}$ as in Lemma 4.3, then {\mathcal{S}}=\big{\{}S_{i},\overline{S_{i}}\mid i\in[1,t]\big{\}} is a $k$-sun system of $\Gamma[2]+2u$. In particular, if \mathcal{C}=Orb\big{(}C_{1}\big{)}, then Orb(S_{1})\ \cup\ Orb\big{(}\overline{S_{1}}\big{)} is a $k$-sun system of $\Gamma[2]+2u$.

Proof.

By assumption $\Gamma+u=\oplus_{i=1}^{t}C_{i}$, where each $C_{i}$ is a $k$-cycle. Also, by Lemma 4.2, we have that $\Gamma[2]+2u=(\Gamma+u)[2]=\oplus_{i=1}^{t}C_{i}[2]$. Since $C_{i}[2]=S_{i}\oplus\overline{S_{i}}$ by Lemma 4.3, then ${\mathcal{S}}$ is a $k$-sun system of $\Gamma[2]+2u$.

The second part easily follows by noticing that if $C_{i}=\tau_{g}(C_{1})$ for some $g\in\mathbb{Z}_{M}$, then C_{i}[2]=\tau_{g}(C_{1}[2])=\tau_{g}\big{(}S_{1}\big{)}\oplus\tau_{g}\big{(}\overline{S_{1}}\big{)}. ∎

The following lemma describes the general method we use to construct $k$-sun systems of $K_{4k}+n$. We point out that throughout the rest of this section we take $V(K_{2k})=\mathbb{Z}_{k}\times\{0,1\}$ and $V(K_{4k})=\mathbb{Z}_{k}\times[0,3]$.

Lemma 4.6**.**

Let $K_{2k}=\Gamma_{1}\oplus\Gamma_{2}$ with $V(\Gamma_{1})=V(\Gamma_{2})=V(K_{2k})$. If $\Gamma_{1}+w_{1}$ has a $k$-cycle system and $\Gamma_{2}^{*}[2]+w_{2}$ has a $k$-sun system, then $K_{4k}+(2w_{1}+w_{2})$ has a $k$-sun system.

Proof.

The result follows by Lemma 4.2. In fact, noting that $K_{4k}=K_{2k}[2]\oplus I$, where I=\big{\{}\{z,\overline{z}\}\mid z\in\mathbb{Z}_{k}\times\{0,1\}\big{\}}, we have that

[TABLE]

The result then follows by Lemma 4.5. ∎

We are now ready to prove the main result of this section, Theorem 4.1. The case $k\equiv 1\pmod{4}$ is proven in Theorem 4.7, while the case $k\equiv 3\pmod{4}$ is dealt with in Theorems 4.9, 4.10, 4.11 and 4.12.

Theorem 4.7**.**

If $k\equiv 1\pmod{4}\geq 9$ and $n\equiv 0,1\pmod{4}$ with $2k<n<10k$, then there exists a $k$-sun system of $K_{4k}+n$.

Proof.

Let $n=2(q\ell+r)+\nu$ with $1\leq r\leq\ell$ and $\nu\in\{2,3\}$. Note that $\ell\geq 4$ is even and $r$ is odd, since $n\equiv 0,1\pmod{4}\geq 9$ and $k\equiv 1\pmod{4}$. Considering also that $2k<n<10k$, we have that $2\leq q\leq 10\leq k+2r-1$. Furthermore, let V(K_{4k}+n)=\big{(}\mathbb{Z}_{k}\times[0,3]\big{)}\cup\{\infty_{h}\mid h\in\mathbb{Z}_{n-\nu}\}\cup\{\infty^{\prime}_{1},\infty^{\prime}_{2},\infty^{\prime}_{\nu}\}.

We start decomposing $K_{2k}$ into the following two graphs:

[TABLE]

We notice that $\Gamma_{1}$ further decomposes into the following graphs:

[TABLE]

each of which decomposes into $k$-cycles by Lemmas 3.1 and 3.2; hence $\Gamma_{1}$ has a $k$-cycle system $\{C_{1},C_{2},\ldots,C_{\gamma}\}$, where $\gamma=k+2r-2$. Note that this system is non-empty, since $1\leq q-1\leq\gamma$. Without loss of generality, we can assume that each cycle $C_{i}$ has order $2k$ and

[TABLE]

Now set $\Omega_{1}=\Gamma_{1}\setminus C_{1}$ and $\Omega_{2}=\Gamma_{2}\oplus C_{1}$. Letting $w_{1}=(q-2)\ell=\sum_{j=2}^{\gamma}w_{1,j}$, where $w_{1,j}=\ell$ when $j<q$, and $w_{1,j}=0$ otherwise, by Lemma 4.2 we have that $\Omega_{1}+w_{1}=\oplus_{i=2}^{\gamma}(C_{i}+w_{1,i})$. Therefore, $\Omega_{1}+w_{1}$ has a $k$-cycle system, since each $C_{i}+w_{1,i}$ decomposes into $k$-cycles by Lemma 3.4. Setting $w_{2}=n-2w_{1}=2(2\ell+r)+\nu$ and considering that $K_{2k}=\Gamma_{1}\oplus\Gamma_{2}=\Omega_{1}\oplus\Omega_{2}$, by Lemma 4.6 it is left to show that $\Omega_{2}^{*}[2]+w_{2}$ has a $k$-sun system.

Set $\Gamma_{3}=C_{1}$, and recall that $\Omega_{2}^{*}[2]=\Omega_{2}[2]\oplus I=\Gamma_{2}[2]\oplus\Gamma_{3}[2]\oplus I$, where $I$ denotes the $1$-factor \big{\{}\{z,\overline{z}\}\mid z\in\mathbb{Z}_{k}\times\{0,1\}\big{\}} of $K_{4k}$. Hence,

[TABLE]

by Lemma 4.2. Clearly, $\Gamma_{2}=\Gamma_{2,1}\oplus\Gamma_{2,2}$ where \Gamma_{2,1}=\big{\langle}\{1\},[0,k-2r-3],\{1\}\big{\rangle} and \Gamma_{2,2}=\big{\langle}\varnothing,\varnothing,\{\ell\}\big{\rangle}, hence $\Gamma_{2}+(\ell+r)=(\Gamma_{2,1}+r)\oplus(\Gamma_{2,2}+\ell)$. By Lemmas 3.3 and 3.4, there exist a $k$-cycle $A=(x_{1},x_{2},y_{3},y_{4},a_{5},\ldots,a_{k})$ of $\Gamma_{2,1}+r$ and a $k$-cycle $B=(y_{1},y_{2},b_{3},\ldots,b_{k})$ of $\Gamma_{2,2}+\ell$ satisfying the following properties:

[TABLE]

Furthermore, denoted by $(c_{1},c_{2},\ldots,c_{k})$ the cycle in $\Gamma_{3}$, Lemma 3.4 guarantees that

[TABLE]

Let ${\mathcal{S}}=\{S_{1},S_{2},S_{3},S_{4}\}$ and ${\mathcal{S}}^{\prime}=\{S_{3+2j},S_{4+2j}\mid j\in[1,k]\}$, where

[TABLE]

By Lemma 4.5 we have that $\bigcup_{S\in{\mathcal{S}}}Orb(S)$ is a $k$-sun system of \big{(}\Gamma_{2}+(\ell+r)\big{)}[2], and ${\mathcal{S}}^{\prime}$ is a $k$-sun system of \big{(}\Gamma_{3}+\ell\big{)}[2]. It follows by (3) that $\bigcup_{S\in{\mathcal{S}}}Orb(S)\ \cup\ {\mathcal{S}}^{\prime}$ decomposes $(\Omega_{2}^{*}[2]+w_{2})\setminus(I+\nu)$.

To construct a $k$-sun system of $\Omega_{2}^{*}[2]+w_{2}$, we first modify the $k$-suns in ${\mathcal{S}}\ \cup\ {\mathcal{S}}^{\prime}$ by replacing some of their vertices with $\infty^{\prime}_{1},\infty^{\prime}_{2}$, and possibly $\infty^{\prime}_{3}$ when $\nu=3$. More precisely, following Table 1, we obtain $T_{i}$ from $S_{i}$ by replacing the ordered set $V_{i}$ of vertices of $S_{i}$ with $V_{i}^{\prime}$. This yields a set $M_{i}$ of ‘missing’ edges no longer covered by $T_{i}$ after this substitution, but replaced by those in $N_{i}$, namely

[TABLE]

We point out that $T_{3+2j}=S_{3+2j}$, and $T_{4+2j}=S_{4+2j}$ when $\nu=2$, for every $j\in[1,k]$. The remaining graphs $T_{i}$ are explicitly given below, where the elements in bold are the replaced vertices.

[TABLE]

We notice that \displaystyle\bigcup_{i=1}^{4}Dev(N_{i})\cup\bigcup_{i=5}^{2k+4}N_{i}=\big{\{}\{\infty^{\prime}_{j},x\}\mid j\in[1,\nu],x\in\mathbb{Z}_{k}\times[0,3]\big{\}}. We finally build the following $2\nu+1$ graphs:

[TABLE]

By recalling (2) and (4)–(6), it is not difficult to check that $G_{1},G_{2},\ldots,G_{2\nu+1}$ are $k$-suns. Furthermore,

[TABLE]

where, we recall, $I$ denotes the $1$-factor \big{\{}\{z,\overline{z}\}\mid z\in\mathbb{Z}_{k}\times\{0,1\}\big{\}} of $K_{4k}$. Therefore, $\bigcup_{i=1}^{4}Orb(T_{i})\ \cup\ \{T_{5},T_{6},\ldots,T_{2k+4}\}\ \cup\ \{G_{1},G_{2},\ldots,G_{2\nu+1}\}$ is a $k$-sun system of $\Omega^{*}_{2}[2]+w_{2}$, and this concludes the proof. ∎

Example 4.8**.**

By following the proof of Theorem 4.7, we construct a $k$-sun system of $K_{4k}+n$ when $(k,n)=(9,21)$; hence $(\ell,q,r,\nu)=(4,2,1,3)$.

The graphs \Gamma_{1}=\big{\langle}[2,4],[5,8],[2,3]\big{\rangle} and \Gamma_{2}=\big{\langle}\{1\},[0,4],\{1,4\}\big{\rangle} decompose the complete graph $K_{18}$ with vertex-set $\mathbb{Z}_{9}\times\{0,1\}$. Also $\Gamma_{1}$ decomposes into the following $9$-cycles of order $18$, where $i=0,1$:

[TABLE]

Clearly, $K_{18}=\Omega_{1}\oplus\Omega_{2}$, where $\Omega_{1}=\Gamma_{1}\setminus{C_{1}}$ and $\Omega_{2}=\Gamma_{2}\oplus C_{1}$.

Let $V(K_{36})=\mathbb{Z}_{9}\times[0,3]$, and denote by $I$ the $1$-factor of $K_{36}$ containing all edges of the form $\{(a,i),(a,i+2)\}$, with $a\in\mathbb{Z}_{9}$ and $i\in\{0,1\}$. Then,

[TABLE]

Considering that $(\Omega_{2}+9)[2]=\Omega_{2}[2]+18$, we have

[TABLE]

Since the set $\{\sigma(C_{i}),\overline{\sigma(C_{i})}\mid i\in[2,9]\}$ is a $9$-sun system of $\Omega_{1}[2]$, it is left to build a $9$-sun system of $\Omega^{*}_{2}[2]+21=(\Omega_{2}[2]+18)\oplus(I+3)$.

We start by decomposing $\Omega_{2}+9$ into $9$-cycles. Since $\Omega_{2}=\Gamma_{2,1}\oplus\Gamma_{2,2}\oplus\Gamma_{3}$ with \Gamma_{2,1}=\big{\langle}\{1\},[0,4],\{1\}\big{\rangle}, \Gamma_{2,2}=\big{\langle}\varnothing,\varnothing,\{4\}\big{\rangle} and $\Gamma_{3}=C_{1}$, then

[TABLE]

Let $A=(x_{1},x_{2},y_{3},y_{4},a_{5},\ldots,a_{9})$ and $B=(y_{1},y_{2},b_{3},\ldots,b_{9})$ be the $9$-cycles defined as follows:

[TABLE]

One can easily check that $Orb(A)$ (resp., $Orb(B)$) decomposes $\Gamma_{2,1}+1$ (resp., $\Gamma_{2,2}+4$). Also, for every edge $\{c_{j},c_{j+1}\}$ of $C_{1}$, with $j\in[1,9]$ and $c_{10}=c_{1}$, we construct the cycle $F_{j}=(c_{j},c_{j+1},f_{j,3},f_{j,4},\ldots,f_{j,9})$, where

[TABLE]

One can check that $\{F_{1},F_{2},\ldots,F_{9}\}$ is a $9$-cycle system of $\Gamma_{3}+4$. Therefore, $\mathcal{U}_{1}=Orb(A)\ \cup\ Orb(B)\ \cup\ \{F_{1},F_{2},\ldots,F_{9}\}$ provides a $9$-cycle system of $\Omega_{2}+9$. Since the set $\{C[2]\mid C\in\mathcal{U}_{1}\}$ decomposes $(\Omega_{2}+9)[2]$, and each $C[2]$ decomposes into two $9$-suns, we can easily obtain a $9$-sun system of $(\Omega_{2}+9)[2]$. Indeed, letting

[TABLE]

we have that $A[2]=S_{1}\oplus S_{2}$, $B[2]=S_{3}\oplus S_{4}$, and $F_{j}[2]=S_{3+2j}\oplus S_{4+2j}$, for every $j\in[1,9]$. Therefore $\mathcal{U}_{2}=\bigcup_{i=1}^{4}Orb(S_{i})\cup\{S_{5},S_{6},\ldots,S_{22}\}$ is a $9$-sun system of $\Omega_{2}[2]+18$.

We finally use $\mathcal{U}_{2}$ to build a $9$-sun system of $\Omega^{*}_{2}[2]+21=(\Omega_{2}[2]+18)\oplus(I+3)$. By replacing the vertices of each $S_{i}$, as outlined in Table 1, we obtain the $9$-sun $T_{i}$. The new 22 graphs, $T_{1},T_{2},\ldots,T_{22}$, are built in such a way that

[TABLE]

This way we obtain a $9$-sun system of $\Omega^{*}_{2}[2]+21$, and hence the desired $9$-sun system of $K_{36}+21$.

Theorem 4.9**.**

Let $k\equiv 3\pmod{4}\geq 7$ and $n\equiv 0,1\pmod{4}$ with $2k<n<10k$. If $n\not\equiv 2,3\pmod{k-1}$ and $\left\lfloor\frac{n-4}{k-1}\right\rfloor$ is even, then there exists a $k$-sun system of $K_{4k}+n$ except possibly when $(k,n)\in\{(7,64),(7,65)\}$.

Proof.

First, $k\equiv 3\pmod{4}\geq 7$ implies that $\ell\geq 3$ is odd. Now, let $n=2(q\ell+r)+\nu$ with $1\leq r\leq\ell$ and $\nu\in\{2,3\}$. Note that $q=\left\lfloor\frac{n-4}{k-1}\right\rfloor$, hence $q$ is even. Also, since $2k<n<10k$, we have $2\leq q\leq 10$. By $q$ even and $n\equiv 0,1\pmod{4}$ it follows that $r$ is odd, and $n\not\equiv 2,3\pmod{k-1}$ implies that $r\neq\ell$. To sum up,

[TABLE]

As in the previous theorem, let V(K_{4k}+n)=\big{(}\mathbb{Z}_{k}\times[0,3]\big{)}\cup\{\infty_{h}\mid h\in\mathbb{Z}_{n-\nu}\}\cup\{\infty^{\prime}_{1},\infty^{\prime}_{2},\infty^{\prime}_{\nu}\}.

We split the proof into two cases.

Case 1) $q\leq 2r+4$. We start decomposing $K_{2k}$ into the following two graphs:

[TABLE]

Since $q\leq 2r+4$, the graph $\Gamma_{1}$ can be further decomposed into the following graphs:

[TABLE]

[TABLE]

The first two graphs have a $k$-cycle system by Lemmas 3.2 and 3.1, while $\Gamma_{1,3}$ decomposes into $(q-1)$ $1$-factors, say $J_{1},J_{2},\ldots,J_{q-1}$. Setting $w_{1}=(q-1)\ell$, by Lemma 4.2 we have that:

[TABLE]

Hence $\Gamma_{1}+(q-1)\ell$ has a $k$-cycle system since each $J_{i}+\ell$ decomposes into $k$-cycles by Lemma 3.4.

Letting $w_{2}=n-2w_{1}=2(\ell+r)+\nu$ and recalling that $K_{2k}=\Gamma_{1}\oplus\Gamma_{2}$, by Lemma 4.6 it remains to construct a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}.$ We start decomposing $\Gamma_{2}$ into the following graphs:

[TABLE]

Recalling that $\Gamma^{*}_{2}[2]=\Gamma_{2}[2]\oplus I$, where $I$ denotes the $1$-factor \big{\{}\{z,\overline{z}\}\mid z\in\mathbb{Z}_{k}\times\{0,1\}\big{\}} of $K_{4k}$, by Lemma 4.2 we have that

[TABLE]

By Lemmas 3.3 and 3.4 there exist a $k$-cycle $A=(x_{1},x_{2},x_{3},y_{4},y_{5},y_{6},a_{7},\ldots,a_{k})$ of $\Gamma_{2,0}+r$ and a $k$-cycle $B=(y,x,b_{3},\ldots,b_{k})$ of $\Gamma_{2,1}+\ell$, satisfying the following properties:

[TABLE]

Set $A^{\prime}=(x_{1},\overline{x_{2}},x_{3},y_{4},\overline{y_{5}},y_{6},a_{7},\ldots,a_{k})$ and $B^{\prime}=(y,\overline{x},b_{3},\ldots,b_{k})$ and let ${\mathcal{S}}=\{\sigma(A^{\prime}),\overline{\sigma(A^{\prime})},\sigma(B^{\prime}),\overline{\sigma(B^{\prime})}\}$. By Lemma 4.5, we have that $\bigcup_{S\in{\mathcal{S}}}Orb(S)$ is a $k$-sun system of \big{(}\Gamma_{2}+(\ell+r)\big{)}[2]=\Gamma_{2}[2]+2(\ell+r)=(\Gamma_{2}^{*}[2]+w_{2})\setminus(I+\nu).

To construct a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$ we proceed as in Theorem 4.7. We modify the graphs in ${\mathcal{S}}$ and obtain four $k$-suns $T_{1},T_{2},T_{3},T_{4}$ whose translates between them cover all edges incident with $\infty^{\prime}_{1},\infty^{\prime}_{2}$, and possibly $\infty^{\prime}_{3}$ when $\nu=3$. Then we construct further $2\nu+1$ $k$-suns $G_{1},\ldots,G_{2\nu+1}$ to cover the missing edges. The reader can check that $\bigcup_{i=1}^{4}Orb(T_{i})\cup\{G_{1},\ldots,G_{2\nu+1}\}$ is a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$.

The graphs $T_{i}$ are the following, where the elements in bold are the replaced vertices:

[TABLE]

The graphs $G_{i}$, for $i=[1,2\nu+1]$, are so defined:

[TABLE]

Case 2) $q\geq 2r+6$. Note that this implies $r=1$ and $q=8,10$. As before $K_{2k}=\Gamma_{1}\oplus\Gamma_{2}$ where

[TABLE]

Since $(k,n)\neq(7,64),(7,65)$ then $(\ell,q)\neq(3,10)$, hence the graph $\Gamma_{1}$ can be decomposed into the following graphs:

[TABLE]

[TABLE]

The graph $\Gamma_{1,1}$ decomposes into five $1$-factors $J_{1},\ldots,J_{5}$, while by Lemma 3.5 $\Gamma_{1,2}$ decomposes into $(q-5)$ $1$-factors $J^{\prime}_{1},\ldots,J^{\prime}_{q-5}$. Letting $w_{1}=q\ell$, by Lemma 4.2 we have that

[TABLE]

By Lemmas 3.4 and 3.1, each $J_{i}+\ell$, each $J^{\prime}_{i}+\ell$ and $\Gamma_{1,3}$ decompose into $k$-cycles. Hence $\Gamma_{1}+q\ell$ has a $k$-cycle system. Let now $w_{2}=n-2w_{1}=2+\nu$. Note that a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$ can be obtained as in Case 1, where $\Gamma_{2,1}$ is empty. ∎

Theorem 4.10**.**

Let $k\equiv 3\pmod{4}\geq 11$ and $n\equiv 0,1\pmod{4}$ with $2k<n<10k$. If $\left\lfloor\frac{n-4}{k-1}\right\rfloor$ is even, and $n\equiv 2,3\pmod{k-1}$, then there is a $k$-sun system of $K_{4k}+n$, except possibly when $(k,n)\in\{(11,112),$ $(11,113)\}$.

Proof.

Let $n=2(q\ell+r)+\nu$ with $1\leq r\leq\ell$ and $\nu\in\{2,3\}$. Clearly, $q=\left\lfloor\frac{n-4}{k-1}\right\rfloor$, hence $q$ is even. Since $k\geq 11$, $2k<n<10k$ and $n\equiv 2,3\pmod{2\ell}$, we have that

[TABLE]

As before, let V(K_{4k}+n)=\big{(}\mathbb{Z}_{k}\times[0,3]\big{)}\cup\{\infty_{h}\mid h\in\mathbb{Z}_{n-\nu}\}\cup\{\infty^{\prime}_{1},\infty^{\prime}_{2},\infty^{\prime}_{\nu}\}.

We start decomposing $K_{2k}$ into the following two graphs:

[TABLE]

If $q=2,4$, $\Gamma_{1}$ can be further decomposed into

[TABLE]

[TABLE]

The graph $\Gamma_{1,1}$ decomposes into $q$ $1$-factors, say $J_{1},\ldots,J_{q}$. Letting $w_{1}=q\ell$, by Lemma 4.2 we have that

[TABLE]

Lemmas 3.4, 3.2 and 3.1 guarantee that each $J_{i}+\ell$, $\Gamma_{1,2}$ and $\Gamma_{1,3}$ decompose into $k$-cycles, hence $\Gamma_{1}+w_{1}$ has a $k$-cycle system. Suppose now $q\geq 6$. By $(k,n)\not\in\{(11,112),(11,113)\}$, we have $(\ell,q)\neq(5,10)$. In this case $\Gamma_{1}$ can be further decomposed into

[TABLE]

[TABLE]

The graph $\Gamma_{1,1}$ can be decomposed into three $1$-factors say $J_{1},J_{2},J_{3}$, also by Lemma 3.5 the graph $\Gamma_{1,2}$ can be decomposed into $(q-3)$ $1$-factors say $J^{\prime}_{1},\ldots,J^{\prime}_{q-3}$. Set again $w_{1}=q\ell$, by Lemma 4.2 we have that

[TABLE]

By Lemmas 3.4 and 3.1 we have that each $J_{i}+\ell$, each $J^{\prime}_{j}+\ell$ and $\Gamma_{1,3}$ decompose into $k$-cycles, hence $\Gamma_{1}+w_{1}$ has a $k$-cycle system. Hence for any value of $q$ we have proved that $\Gamma_{1}+w_{1}$ has a $k$-cycle system.

Now, setting $w_{2}=n-2w_{1}=2\ell+\nu$ and recalling that $K_{2k}=\Gamma_{1}\oplus\Gamma_{2}$, by Lemma 4.6 it is left to show that $\Gamma_{2}^{*}[2]+w_{2}$ has a $k$-sun system. Let $r_{1}$ and $r_{2}\geq 2$ be an odd and an even integer, respectively, such that $r_{1}+r_{2}=r=\ell$. Note that $\Gamma_{2}$ can be further decomposed into

[TABLE]

Recalling that $\Gamma^{*}_{2}[2]=\Gamma_{2}[2]\oplus I$, where $I$ denotes the $1$-factor \big{\{}\{z,\overline{z}\}\mid z\in\mathbb{Z}_{k}\times\{0,1\}\big{\}} of $K_{4k}$, by Lemma 4.2 we have that

[TABLE]

By Lemma 3.3 there is a $k$-cycle $A=(y_{1},y_{2},x_{3},x_{4},a_{5},\ldots,a_{k})$ of $\Gamma_{2,1}+r_{1}$ and a $k$-cycle $B=(x_{1},x_{2},y_{3},y_{4},b_{5},\ldots,b_{k})$ of $\Gamma_{2,2}+r_{2}$ such that

[TABLE]

Set $A^{\prime}=(y_{1},\overline{y_{2}},x_{3},\overline{x_{4}},a_{5},\ldots,a_{k})$ and $B^{\prime}=(x_{1},\overline{x_{2}},y_{3},\overline{y_{4}},b_{5},\ldots,b_{k})$. Let ${\mathcal{S}}=\{\sigma(A^{\prime}),\overline{\sigma(A^{\prime})},\sigma(B^{\prime}),\overline{\sigma(B^{\prime})}\}$, by Lemma 4.5, we have that $\bigcup_{S\in{\mathcal{S}}}Orb(S)$ is a $k$-sun system of \big{(}\Gamma_{2}+\ell\big{)}[2]=\Gamma_{2}[2]+2\ell=(\Gamma_{2}^{*}[2]+w_{2})\setminus(I+\nu). To construct a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$, we build a family ${\mathcal{T}}=\{T_{1},T_{2},T_{3},T_{4}\}$ of $k$-suns by modifying the graphs in ${\mathcal{S}}$ so that $\bigcup_{T\in{\mathcal{T}}}Orb(T)$ covers all the edges incident with $\infty^{\prime}_{1},\infty^{\prime}_{2}$, and possibly $\infty^{\prime}_{3}$ when $\nu=3$. We then construct further $(2\nu+1)$ $k$-suns $G_{1},G_{2},\ldots,G_{2\nu+1}$ which cover the remaining edges exactly once. Hence, $\bigcup_{T\in{\mathcal{T}}}Orb(T)\cup\{G_{1},G_{2},\ldots,G_{2\nu+1}\}$ is a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$.

The graphs $T_{1},\ldots,T_{4}$ and $G_{1},\ldots,G_{2\nu+1}$ are the following, where as before the elements in bold are the replaced vertices.

[TABLE]

[TABLE]

By recalling (LABEL:thm:holek=3,a2), it is not difficult to check that the graphs $G_{h}$ are $k$-suns.

∎

Theorem 4.11**.**

*Let $k\equiv 3\pmod{4}\geq 7$ and $n\equiv 0,1\pmod{4}$ with $2k<n<10k$. If $\left\lfloor\frac{n-4}{k-1}\right\rfloor$ is odd and $n\not\equiv 0,1\pmod{k-1}$, then there is a $k$-sun system of $K_{4k}+n$. *

Proof.

Let $n=2(q\ell+r)+\nu$ with $1\leq r\leq\ell$ and $\nu\in\{2,3\}$. Clearly, $q=\lfloor\frac{n-4}{k-1}\rfloor$. Also, we have that $q$ and $\ell\geq 3$ are odd, and $n\equiv 0,1\pmod{4}$; hence $r$ is even. Furthermore, we have that $2\leq q\leq 10$, since by assumption $2k<n<10k$. Considering now the hypothesis that $n\not\equiv 0,1\pmod{2\ell}$, it follows that $r\neq\ell-1$. To sum up,

[TABLE]

As before, let V(K_{4k}+n)=\big{(}\mathbb{Z}_{k}\times[0,3]\big{)}\cup\{\infty_{h}\mid h\in\mathbb{Z}_{n-\nu}\}\cup\{\infty^{\prime}_{1},\infty^{\prime}_{2},\infty^{\prime}_{\nu}\}.

We start decomposing $K_{2k}$ into the following two graphs:

[TABLE]

Considering that $3\leq q\leq 9\leq 2r+5$, the graph $\Gamma_{1}$ can be further decomposed into the following graphs:

[TABLE]

[TABLE]

The first two have a $k$-cycle system by Lemmas 3.1 and 3.2, while $\Gamma_{1,3}$ decomposes into $(q-3)$ $1$-factors, say $J_{1},J_{2},\ldots,J_{q-3}$. Letting $w_{1}=(q-3)\ell$, by Lemma 4.2 we have that

[TABLE]

Therefore, $\Gamma_{1}+w_{1}$ has a $k$-cycle system, since each $J_{i}+\ell$ decomposes into $k$-cycles by Lemma 3.4. Setting $w_{2}=n-2w_{1}=2(3\ell+r)+\nu$ and recalling that $K_{2k}=\Gamma_{1}\oplus\Gamma_{2}$, by Lemma 4.6 it is left to show that $\Gamma_{2}^{*}[2]+w_{2}$ has a $k$-sun system.

We start decomposing $\Gamma_{2}$ into the following graphs:

[TABLE]

Recalling that $\Gamma^{*}_{2}[2]=\Gamma_{2}[2]\oplus I$, where $I$ denotes the $1$-factor \big{\{}\{z,\overline{z}\}\mid z\in\mathbb{Z}_{k}\times\{0,1\}\big{\}} of $K_{4k}$, by Lemma 4.2 we have that

[TABLE]

By Lemmas 3.3 and 3.4 there exist a $k$-cycle $A=(x_{1},x_{2},x_{3},y_{4},y_{5},y_{6},a_{7},\ldots,a_{k})$ of $\Gamma_{2,0}+r$, a $k$-cycle $B_{1}=(x_{1,0},y_{1,1},b_{1,2},\ldots,b_{1,k-1})$ of $\Gamma_{2,1}+\ell$, and a $k$-cycle $B_{i}=(y_{i,0},x_{i,1},b_{i,2},\ldots,b_{i,k-1})$ of $\Gamma_{2,i}+\ell$, for $2\leq i\leq 3$, satisfying the following properties:

[TABLE]

Set $A^{\prime}=(x_{1},\overline{x_{2}},x_{3},\overline{y_{4}},y_{5},\overline{y_{6}},a_{7},a_{8},\ldots,a_{k-1},a_{k})$ and let ${\mathcal{S}}=\{\sigma(A^{\prime}),\overline{\sigma(A^{\prime})}\}\ \cup\ \{\sigma(B_{i}),\overline{\sigma(B_{i})}\mid 1\leq i\leq 3\}$. By Lemma 4.5, we have that $\bigcup_{S\in{\mathcal{S}}}Orb(S)$ is a $k$-sun system of \big{(}\Gamma_{2}+(3\ell+r)\big{)}[2]=\Gamma_{2}[2]+2(3\ell+r)=(\Gamma_{2}^{*}[2]+w_{2})\setminus(I+\nu).

To construct a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$, we build a family ${\mathcal{T}}=\{T_{0},T_{1},\ldots,T_{7}\}$ of $k$-suns by modifying the graphs in ${\mathcal{S}}$ so that $\bigcup_{T\in{\mathcal{T}}}Orb(T)$ covers all the edges incident with $\infty^{\prime}_{1},\infty^{\prime}_{2}$, and possibly $\infty^{\prime}_{3}$ when $\nu=3$. We then construct further $(2\nu+1)$ $k$-suns $G_{1},G_{2},\ldots,G_{2\nu+1}$ which cover the remaining edges exactly once. Hence, $\bigcup_{T\in{\mathcal{T}}}Orb(T)\cup\{G_{1},G_{2},\ldots,G_{2\nu+1}\}$ is a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$.

The graphs $T_{0},\ldots,T_{7}$ and $G_{1},\ldots,G_{2\nu+1}$ are the following, where as before the elements in bold are the replaced vertices.

[TABLE]

[TABLE]

By recalling (LABEL:thm:holek=3,b1:cond2)–(11), it is not difficult to check that the graphs $G_{h}$ are $k$-suns. ∎

Theorem 4.12**.**

Let $k\equiv 3\pmod{4}\geq 7$ and $n\equiv 0,1\pmod{4}$ with $2k<n<10k$. If $\left\lfloor\frac{n-4}{k-1}\right\rfloor$ is odd, and $n\equiv 0,1\pmod{k-1}$, then there is a $k$-sun system of $K_{4k}+n$ except possibly when $(k,n)\in\{(11,100),(11,101)\}$.

Proof.

Let $n=2(q\ell+r)+\nu$ with $1\leq r\leq\ell$ and $\nu\in\{2,3\}$. Reasoning as in the proof of Theorem 4.11 and considering that $n\equiv 0,1\pmod{2\ell}$ and $(k,n)\not\in\{(11,100),(11,101)\}$, we have that

[TABLE]

As before, let V(K_{4k}+n)=\big{(}\mathbb{Z}_{k}\times[0,3]\big{)}\cup\{\infty_{h}\mid h\in\mathbb{Z}_{n-\nu}\}\cup\{\infty^{\prime}_{1},\infty^{\prime}_{2},\infty^{\prime}_{\nu}\}.

We start decomposing $K_{2k}$ into the following two graphs

[TABLE]

Considering (12), we can further decompose $\Gamma_{1}$ into the following two graphs:

[TABLE]

By Lemma 3.5, the graph $\Gamma_{1,1}$ decomposes into $q$ $1$-factors, say $J_{1},J_{2},\ldots,J_{q}$. Letting $w_{1}=q\ell$, by Lemma 4.2 we have that

[TABLE]

Lemmas 3.4 and 3.1 guarantee that each $J_{i}+\ell$ and $\Gamma_{1,2}$ decompose into $k$-cycles, hence $\Gamma_{1}+w_{1}$ has a $k$-cycle system. Let $r_{1}$ and $r_{2}$ be odd positive integers such that $r=\ell-1=r_{1}+r_{2}$. Then, setting $w_{2}=n-2w_{1}=2(r_{1}+r_{2})+\nu$ and recalling that $K_{2k}=\Gamma_{1}\oplus\Gamma_{2}$, by Lemma 4.6 it is left to show that $\Gamma_{2}^{*}[2]+w_{2}$ has a $k$-sun system.

We start decomposing $\Gamma_{2}$ into the following graphs:

[TABLE]

Recalling that $\Gamma^{*}_{2}[2]=\Gamma_{2}[2]\oplus I$, where $I$ denotes the $1$-factor \big{\{}\{z,\overline{z}\}\mid z\in\mathbb{Z}_{k}\times\{0,1\}\big{\}} of $K_{4k}$, by Lemma 4.2 we have that

[TABLE]

By Lemma 3.3 there is a $k$-cycle $A=(y_{1},y_{2},x_{3},x_{4},a_{5},\ldots,a_{k})$ of $\Gamma_{2,1}+r_{1}$ and a $k$-cycle $B=(x_{1},x_{2},y_{3},y_{4},b_{5},\ldots,b_{k})$ of $\Gamma_{2,2}+r_{2}$ such that

[TABLE]

Set $A^{\prime}=(y_{1},\overline{y_{2}},x_{3},\overline{x_{4}},a_{5},\ldots,a_{k})$, $B^{\prime}=(x_{1},\overline{x_{2}},\overline{y_{3}},y_{4},b_{5},\ldots,b_{k})$ and let ${\mathcal{S}}=\{\sigma(A^{\prime}),\overline{\sigma(A^{\prime})},\sigma(B^{\prime}),\overline{\sigma(B^{\prime})}\}$. By Lemma 4.5, we have that $\bigcup_{S\in{\mathcal{S}}}Orb(S)$ is a $k$-sun system of $(\Gamma_{2}^{*}[2]+w_{2})\setminus(I+\nu)$.

To construct a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$, we build a family ${\mathcal{T}}=\{T_{1},T_{2},T_{3},T_{4}\}$ of four $k$-suns, each of which is obtained from a graph in ${\mathcal{S}}$ by replacing some of their vertices with $\infty^{\prime}_{1},\infty^{\prime}_{2}$, and possibly $\infty^{\prime}_{3}$ when $\nu=3$. Then we construct further $(2\nu+1)$ $k$-suns $G_{1},G_{2},\ldots,G_{2\nu+1}$ so that $\bigcup_{T\in{\mathcal{T}}}Orb(T)\ \cup\ \{G_{1},G_{2},\ldots,G_{2\nu+1}\}$ is a $k$-sun system of $\Gamma_{2}^{*}[2]+w_{2}$.

[TABLE]

[TABLE]

By (13), it is not difficult to check that the graphs $G_{h}$ are $k$-suns. ∎

5. It is sufficient to solve $2k<v<6k$

In this section we show that if the necessary conditions in ($\ast$ ‣ 1), for the existence of a $k$-sun system of $K_{v}$, are sufficient for all $v$ satisfying $2k<v<6k$, then they are sufficient for all $v$. In other words, we prove Theorem 1.1.

We start by showing how to construct $k$-sun systems of $K_{g\times h}$ (i.e., the complete multipartite graph with $g$ parts each of size $h$) when $h=4k$.

Theorem 5.1**.**

For any odd integer $k\geq 3$ and any integer $g\geq 3$, there exists a $k$-sun system of $K_{g\times 4k}$.

Proof.

Set $V(K_{g\times 2k})=\mathbb{Z}_{gk}\times[0,1]$ and let $K_{g\times 4k}=K_{g\times 2k}[2]$. In [11, Theorem 2] the authors proved the existence of a $k$-cycle system of $K_{g\times 2k}$. By applying Lemma 4.5 (with $\Gamma=K_{g\times 2k}$ and $u=0$) we obtain the existence of a $k$-sun system of $K_{g\times 4k}$. ∎

The following result exploits Theorem 5.1 and shows how to construct $k$-sun systems of $K_{4kg+n}$, for $g\neq 2$, starting from a $k$-sun system of $K_{4k}+n$ and a $k$-sun system of either $K_{n}$ or $K_{4k+n}$.

Theorem 5.2**.**

Let $k\geq 3$ be an odd integer and assume that both the following conditions hold:

1. (1)

there exists a $k$-sun system of either $K_{n}$ or $K_{4k+n}$; 2. (2)

there exists a $k$-sun system of $K_{4k}+n$.

Then there is a $k$-sun system of $K_{4kg+n}$ for all positive $g\neq 2$.

Proof.

Suppose there exists a $k$-sun system ${\mathcal{S}}_{1}$ of $K_{n}$, also, by (2), there exists a $k$-sun system ${\mathcal{S}}_{2}$ of $K_{4k}+n$. Clearly, ${\mathcal{S}}_{1}\ \cup\ {\mathcal{S}}_{2}$ is a $k$-sun system of $K_{n+4k}=K_{n}\oplus(K_{4k}+n)$. Hence we can suppose $g\geq 3$. Let $V$, $H$ and $G$ be sets of size $n$, $4k$ and $g$, respectively, such that $V\cap(H\times G)=\emptyset$. Let $\mathcal{S}$ be a $k$-sun system of $K_{n}$ (resp., $K_{n+4k}$) with vertex set $V$ (resp., $V\cup(H\times\{x_{0}\})$ for some $x_{0}\in G$). By assumption, for each $x\in G$, there is a $k$-sun system, say ${\mathcal{B}}_{x}$, of $K_{4k}+n$ with vertex set $V\cup(H\times\{x\})$, where $V(K_{4k})=H\times\{x\}$. Also, by Theorem 5.1 there is a $k$-sun system $\mathcal{C}$ of $K_{g\times 4k}$ whose parts are $H\times\{x\}$ with $x\in G$. Hence the $k$-suns of ${\mathcal{B}}_{x}$ with $x\in G$ (resp., $x\in G\setminus\{x_{0}\}$), $\mathcal{S}$ and $\mathcal{C}$ form a $k$-sun system of $K_{n+4kg}$ with vertex set $V\cup(H\times G)$. ∎

We are now ready to prove Theorem 1.1 whose statement is recalled below.

Theorem 1.1. * Let $k\geq 3$ be an odd integer and $v>1$. Conjecture 1 is true if and only if there exists a $k$-sun system of $K_{v}$ for all $v$ satisfying the necessary conditions in ($\ast$ ‣ 1) with $2k<v<6k$.*

Proof.

The existence of $3$-sun systems and $5$-sun systems has been solved in [10] and in [8], respectively. Hence we can suppose $k\geq 7$ and $2k<v<6k$.

We first deal with the case where $(k,v)\neq(7,21)$. By assumption there exists a $k$-sun system of $K_{v}$, which implies $v(v-1)\equiv 0\pmod{4}$, hence Theorem 4.1 guarantees the existence of a $k$-sun system of $K_{4k}+v$. Therefore, by Theorem 5.2 there is a $k$-sun decomposition of $K_{4kg+v}$ whenever $g\neq 2$. To decompose $K_{8k+v}$ into $k$-suns, we first decompose $K_{8k+v}$ into $K_{4k+v}$ and $K_{4k}+(4k+v)$. By Theorem 5.2 (with $g=1$), there is a $k$-sun system of $K_{4k+v}$. Furthermore, Theorem 4.1 guarantees the existence of a $k$-sun system of $K_{4k}+(4k+v)$, except possibly when $(k,4k+v)\in\{(7,56),(7,57),(7,64),(11,100)\}$. Therefore, by Theorem 5.2, there is a $k$-sun decomposition of $K_{8k+v}$ whenever $(k,4k+v)\not\in\{(7,56),(7,57),(7,64),(11,100)\}$. For each of these four cases we construct $k$-sun systems of $K_{8k+v}$ as follows.

If $k=7$ and $4k+v=56$, set $V(K_{84})=\mathbb{Z}_{83}\cup\{\infty\}$. We consider the following $7$-suns

[TABLE]

One can easily check that $\bigcup_{i=1}^{3}Orb_{\mathbb{Z}_{83}}(T_{i})$ is a $7$-sun system of $K_{84}$.

If $k=7$ and $4k+v=57$, set $V(K_{85})=\mathbb{Z}_{85}$. Let $T_{1}$ and $T_{2}$ be defined as above, and let $T^{\prime}_{3}$ be the graph obtained from $T_{3}$ replacing $\infty$ with $60$. It is immediate that $\bigcup_{i=1}^{2}Orb_{\mathbb{Z}_{85}}(T_{i})\cup Orb_{\mathbb{Z}_{85}}(T^{\prime}_{3})$ is a $7$-sun system of $K_{85}$.

If $k=7$ and $4k+v=64$, set $V(K_{92})=(\mathbb{Z}_{7}\times\mathbb{Z}_{13})\cup\{\infty\}$. We consider the following $7$-suns

[TABLE]

[TABLE]

One can easily check that $\bigcup_{i=1}^{3}Orb_{\mathbb{Z}_{7}\times\mathbb{Z}_{13}}(T_{i})\ \cup\ \bigcup_{i=4}^{5}Orb_{\{0\}\times\mathbb{Z}_{13}}(T_{i})$ is a $7$-sun system of $K_{92}$.

If $k=11$ and $4k+v=100$, set $V(K_{144})=(\mathbb{Z}_{11}\times\mathbb{Z}_{13})\cup\{\infty\}$. We consider the following $11$-suns

[TABLE]

[TABLE]

One can check that $\bigcup_{i=1}^{3}Orb_{\mathbb{Z}_{11}\times\mathbb{Z}_{13}}(T_{i})\ \cup\ \bigcup_{i=4}^{6}Orb_{\{0\}\times\mathbb{Z}_{13}}(T_{i})$ is an $11$-sun system of $K_{144}$.

It is left to prove the existence of a $k$-sun system of $K_{4kg+v}$ when $(k,v)=(7,21)$ and for every $g\geq 1$. If $g=1$, a $7$-sun system of $K_{49}$ can be obtained as a particular case of the following construction. Let $p$ be a prime, $q=p^{n}\equiv 1\pmod{4}$ and $r$ be a primitive root of $\mathbb{F}_{q}$. Setting $S=Dev_{\langle r\rangle}(0\sim r\sim r+1)$ where $\langle r\rangle=\{jr\mid 1\leq j\leq p\}$, we have that $\bigcup_{i=0}^{\frac{q-5}{4}}Orb_{\mathbb{F}_{q}}(r^{2i}S)$ is a $p$-sun system of $K_{q}$.

If $g\geq 2$, we notice that $K_{28g+21}=K_{28(g-1)+49}$. Considering the $7$-sun system of $K_{49}$ just built, and recalling that by Theorem 4.1 there is a $7$-sun system of $K_{28}+49$, then Theorem 5.2 guarantees the existence of a $7$-sun system of $K_{28(g-1)+49}$ whenever $g\neq 3$. When $g=3$, a $7$-sun system of $K_{105}$ is constructed as follows. Set $V(K_{105})=\mathbb{Z}_{7}\times\mathbb{Z}_{15}$. Let $S_{i,j}$ and $T$ be the $7$-suns defined below, where $(i,j)\in X=([1,3]\times[1,7])\setminus\{(1,3),(1,6)\}$:

[TABLE]

One can check that $\displaystyle\bigcup_{(i,j)\in X}Orb_{\{0\}\times\mathbb{Z}_{15}}(S_{i,j})\ \cup\ Orb_{\mathbb{Z}_{7}\times\mathbb{Z}_{15}}(T)$ is a $7$-sun system of $K_{105}$. ∎

6. Construction of $p$-sun systems, $p$ prime

In this section we prove Theorem 1.2. Clearly in view of Theorem 1.1 it is sufficient to construct a $p$-sun system of $K_{v}$ for any admissible $v$ with $2p<v<6p$. Hence, we are going to prove the following result.

Theorem 6.1**.**

Let $p$ be an odd prime and let $v(v-1)\equiv 0\pmod{4p}$ with $2p<v<6p$. Then there exists a $p$-sun system of $K_{v}$.

Since the existence of $p$-sun systems with $p=3,5$ has been proved in [10] and in [8], respectively, here we can assume $p\geq 7$.

It is immediate to see that by the necessary conditions for the existence of a $p$-sun system of $K_{v}$, it follows that $v$ lies in one of the following congruence classes modulo $4p$:

$v\equiv 0,1\pmod{4p}$;

• 2)

$v\equiv p,3p+1\pmod{4p}$ if $p\equiv 1\pmod{4}$;

• 3)

$v\equiv p+1,3p\pmod{4p}$ if $p\equiv 3\pmod{4}$.

If $v\equiv 0,1\pmod{4p}$ we present a direct construction which holds more in general for $p=k$, where $k$ is an odd integer and not necessarily a prime.

Theorem 6.2**.**

For any $k=2t+1\geq 7$ there exists a $k$-sun system of $K_{4k+1}$ and a $k$-sun system of $K_{4k}$.

Proof.

Let $C$ be the $k$-cycle with vertices in $\mathbb{Z}$ so defined:

[TABLE]

Note that the list $D_{1}$ of the positive differences in $\mathbb{Z}$ of $C$ is $D_{1}=[1,2t]\cup\{3t\}$. Consider now the ordered $k$-set $D_{2}=\{d_{1},d_{2},\ldots,d_{k}\}$ so defined:

[TABLE]

Obviously $D_{1}\cup D_{2}=[1,2k].$ Let $\{c_{1},c_{2},\ldots,c_{k}\}$ be the increasing order of the vertices of the cycle $C$ and set $\ell_{r}=c_{r}+d_{r}$ for every $r\in[1,k]$, with $r\neq\frac{t+1}{2}$, and $\ell_{\frac{t+1}{2}}=c_{\frac{t+1}{2}}-d_{\frac{t+1}{2}}$ when $t$ is odd. It is not hard to see that $V=\{c_{1},c_{2},\ldots,c_{k},\ell_{1},\ell_{2},\ldots,\ell_{k}\}$ is a set. Note also that $V\subseteq\{-3t-1\}\cup[-t,5t]\cup\{6t+2\}$.

Let $S$ be the sun obtainable from $C$ by adding the pendant edges $\{c_{i},\ell_{i}\}$ for $i\in[1,k]$. Clearly, $\Delta S=\pm(D_{1}\ \cup\ D_{2})=\pm[1,2k]$. So we can conclude that if we consider the vertices of $S$ as elements of $\mathbb{Z}_{4k+1}$, the vertices are still pairwise distinct and $\Delta S=\mathbb{Z}_{4k+1}\setminus\{0\}$. Then, by applying Corollary 2.2 (with $G=\mathbb{Z}_{4k+1},m=1,w=0$), it follows that $Orb_{\mathbb{Z}_{4k+1}}S$ is a $k$-sun system of $K_{4k+1}$.

Now we construct a $k$-sun system of $K_{4k}$. Let $S$ be defined as above and note that $d_{k}=2k$. Let $S^{*}$ be the sun obtained by $S$ setting $\ell_{k}=\infty$. It is immediate that if we consider the vertices of $S^{*}$ as elements of $\mathbb{Z}_{4k-1}\cup\{\infty\}$, then Corollary 2.2 (with $G=\mathbb{Z}_{4k-1},m=1,w=1$) guarantees that $Orb_{\mathbb{Z}_{4k-1}}S^{*}$ is a $k$-sun system of $K_{4k}$. ∎

Example 6.3**.**

Let $k=2t+1=9$, hence $t=4$. By following the proof of Theorem 6.2, we construct a $9$-sun system of $K_{37}$. Taking $C=(0,-1,1,-2,2,-3,3,-4,8)$, we have that

[TABLE]

Hence $\{\ell_{1},\ell_{2},\ldots,\ell_{9}\}=\{5,7,9,12,14,16,18,20,26\}$ and we obtain the following $9$-sun $S$ with vertices in $\mathbb{Z}_{37}$:

[TABLE]

such that $\Delta S=\mathbb{Z}_{37}\setminus\{0\}$. Therefore, $Orb_{\mathbb{Z}_{37}}S$ is a $9$-sun system of $K_{37}$.

From now on, we assume that $p$ is an odd prime number and denote by $\Sigma$ the following $p$-sun:

[TABLE]

Lemma 6.4**.**

Let $p$ be an odd prime. For any $x,y\in\mathbb{Z}_{p}$ with $x\neq 0$ and any $i,j\in\mathbb{Z}_{m}$ with $i\neq j$ there exists a $p$-sun $S$ such that $\Delta_{ii}S=\pm{x}$, $\Delta_{ij}S=y$, $\Delta_{ji}S=-y$ and $\Delta_{hk}S=\emptyset$ for any $(h,k)\in(\mathbb{Z}_{m}\times\mathbb{Z}_{m})\setminus\{(i,i),(i,j),(j,i)\}$.

Proof.

It is easy to see that $S=Dev_{\mathbb{Z}_{p}\times\{0\}}((0,i)\sim(x,i)\sim(y+x,j))$ is the required $p$-sun. ∎

We will call such a $p$-sun a sun of type $(i,j)$. For the following it is important to note that if $S$ is a $p$-sun of type $(i,j)$, then $|\Delta_{ii}S|=2$, $|\Delta_{jj}S|=0$ and $|\Delta_{ij}S|=|\Delta_{ji}S|=1$.

The following two propositions provide us $p$-sun systems of $K_{mp+1}$ whenever $m\in\{3,5\}$ and $p\equiv m-2\pmod{4}$.

Proposition 6.5**.**

Let $p\equiv 1\pmod{4}\geq 13$ be a prime. Then there exists a $p$-sun system of $K_{3p+1}$.

Proof.

We have to distinguish two cases according to the congruence of $p$ modulo 12.

Case 1. Let $p\equiv 1\pmod{12}$.

If $p=13$, we construct a $13$-sun system of $K_{40}$ as follows. Let $S$ be the following $13$-sun whose vertices are labelled with elements of $(\mathbb{Z}_{13}\times\mathbb{Z}_{3})\cup\{\infty\}$:

[TABLE]

We have:

[TABLE]

Now it remains to construct a set ${\mathcal{T}}$ of edge-disjoint $13$-suns such that

[TABLE]

In order to do this it is sufficient to take, ${\mathcal{T}}=\{T_{01}^{i}\mid i\in[1,4]\}\cup\{T_{02}^{i}\mid i\in[1,2]\}\cup\{T_{10}^{i}\mid i\in[1,3]\}\cup\{T_{12}^{i}\mid i\in[1,2]\}\cup\{T_{20}^{i}\mid i\in[1,3]\}\cup\{T_{21}^{i}\mid i\in[1,3]\}$, where:

[TABLE]

We have that ${\mathcal{T}}\cup Orb_{\mathbb{Z}_{13}\times\{0\}}S$ is a $13$-sun system of $K_{40}$.

Suppose now that $p\geq 37$. We proceed in a very similar way to the previous case. Let $r$ be a primitive root of $\mathbb{Z}_{p}$. Consider the $((\mathbb{Z}_{p}\times\mathbb{Z}_{3})\cup\{\infty\})$-labeling $B$ of $\Sigma$ so defined:

[TABLE]

except for $\frac{p-9}{4}$ values of $i\equiv 1\pmod{3}$ for which we set $B(\ell_{i})=(r^{i+1},i)$. Letting $S=B(\Sigma)$, it is immediate that the labels of the vertices of $S$ are pairwise distinct. Note that

[TABLE]

[TABLE]

Hence, reasoning as in the previous case, we have to construct a set ${\mathcal{T}}$ of $p$-suns such that if $i\neq j$ then $\Delta_{ij}{\mathcal{T}}=\mathbb{Z}_{p}\setminus\Delta_{ij}S$ is a set and also $\Delta_{ii}{\mathcal{T}}=\mathbb{Z}^{*}_{p}\setminus\Delta_{ii}S$ is a set. In particular, this implies that for any $T,T^{\prime}\in{\mathcal{T}}$ we have $\Delta_{ij}T\cap\Delta_{ij}T^{\prime}=\emptyset$ and that $|\Delta_{00}{\mathcal{T}}|=|\Delta_{22}{\mathcal{T}}|=p-1$, $|\Delta_{11}{\mathcal{T}}|=\frac{p+7}{2}$, $|\Delta_{ij}{\mathcal{T}}|=\frac{p+2}{3}$ for $(i,j)\in\{(0,2),(1,2),(2,0),(2,1)\}$, and $|\Delta_{01}{\mathcal{T}}|=|\Delta_{10}{\mathcal{T}}|=\frac{7p-7}{12}$. In order to do this it is sufficient to take ${\mathcal{T}}$ as a set consisting of $\frac{p-1}{2}$ suns of type $(0,1)$, $\frac{p-1}{12}$ suns of type $(1,0)$, $\frac{p+11}{6}$ suns of type $(1,2)$, $\frac{p+2}{3}$ suns of type $(2,0)$, $\frac{p-7}{6}$ suns of type $(2,1)$, which exist in view of Lemma 6.4. We have that $Orb_{\mathbb{Z}_{p}\times\{0\}}S\cup{\mathcal{T}}$ is a $p$-sun system of $K_{3p+1}$.

Case 2. Let $p\equiv 5\pmod{12}$. Let $r$ be a primitive root of $\mathbb{Z}_{p}$. Consider the $((\mathbb{Z}_{p}\times\mathbb{Z}_{3})\cup\{\infty\})$-labeling $B$ of $\Sigma$ so defined:

[TABLE]

except for $\frac{p-17}{6}$ values of $i\equiv 0\pmod{3}$ with $i\in\left[3,\frac{p-1}{2}\right]$ for which we set $B(\ell_{i})=(r^{i-1},i)$ and $\frac{p-5}{12}$ values of $i\equiv 0\pmod{3}$ with $i\in\left[\frac{p+1}{2},p-5\right]$ for which we set $B(\ell_{i})=(r^{i+1},i)$. Letting $S=B(\Sigma)$, it is easy to see that the labels of the vertices of $S$ are pairwise distinct. Note that

[TABLE]

Hence, we have to construct a set ${\mathcal{T}}$ of $p$-suns such that $|\Delta_{11}{\mathcal{T}}|=|\Delta_{22}{\mathcal{T}}|=p-1$, $|\Delta_{00}{\mathcal{T}}|=\frac{p+7}{2}$, $|\Delta_{01}{\mathcal{T}}|=|\Delta_{10}{\mathcal{T}}|=\frac{p-1}{2}$, $|\Delta_{02}{\mathcal{T}}|=|\Delta_{20}{\mathcal{T}}|=\frac{5p-1}{12}$, and $|\Delta_{12}{\mathcal{T}}|=|\Delta_{21}{\mathcal{T}}|=\frac{p+4}{3}$. In order to do this it is sufficient to take ${\mathcal{T}}$ as a set consisting of $\frac{p+7}{4}$ suns of type $(0,1)$, $\frac{p-9}{4}$ suns of type $(1,0)$, $\frac{p+7}{4}$ suns of type $(1,2)$, $\frac{5p-1}{12}$ suns of type $(2,0)$, and $\frac{p-5}{12}$ suns of type $(2,1)$ which exist in view of Lemma 6.4. We have that $Orb_{\mathbb{Z}_{p}}S\cup{\mathcal{T}}$ is a $p$-sun system of $K_{3p+1}$. ∎

Proposition 6.6**.**

For any prime $p\equiv 3\pmod{4}$ there exists a $p$-sun system of $K_{5p+1}$.

Proof.

Set $p=4n+3$, and let $Y=[1,n]$ and $X=[n+1,2n+1]$. Consider the following $(\mathbb{Z}_{p}\times\mathbb{Z}_{5})\cup\{\infty\}$-labeling $B$ of $\Sigma$ defined as follows:

[TABLE]

One can directly check that the vertices of $S=B(\Sigma)$ are pairwise distinct. Also, it is not hard to verify that $\Delta S$ does not have repetitions and that its complement in $(\mathbb{Z}_{p}\times\mathbb{Z}_{5})\setminus\{(0,0)\}$ is the set

[TABLE]

Clearly, $D$ can be partitioned into $n+1$ quadruples of the form $D_{x}=\{\pm(2x,0),$ $\pm(r_{x},s_{x})\}$ with $x\in X$ and $s_{x}\neq 0$. Letting

[TABLE]

for $x\in X$, it is clear that $\Delta S_{x}=D_{x}$, hence $\Delta\{S_{x}\mid x\in X\}=D$. Therefore, Corollary 2.2 guarantees that $\bigcup_{x\in X}Orb_{\{0\}\times\mathbb{Z}_{5}}(S_{x})\ \cup\ Orb_{\mathbb{Z}_{p}\times\mathbb{Z}_{5}}(S)$ is a $p$-sun system of $K_{5p+1}$. ∎

Example 6.7**.**

Here, we construct a $7$-sun system of $K_{36}$ following the proof of Proposition 6.6. In this case, $Y=\{1\}$ and $X=\{2,3\}$. Now consider the $7$-sun $S$ defined below, whose vertices lie in $(\mathbb{Z}_{7}\times\mathbb{Z}_{5})\cup\{\infty\}$:

[TABLE]

We have

[TABLE]

Hence $\Delta S$ does not have repetitions and its complement in $(\mathbb{Z}_{7}\times\mathbb{Z}_{5})\setminus\{(0,0)\}$ is the set

[TABLE]

Now it is sufficient to take

[TABLE]

One can check that $\bigcup_{x\in X}Orb_{\{0\}\times\mathbb{Z}_{5}}(S_{x})\ \cup\ Orb_{\mathbb{Z}_{7}\times\mathbb{Z}_{5}}S$ is a $7$-sun system of $K_{36}$.

We finally construct $p$-sun systems of $K_{mp}$ whenever $p\equiv m\pmod{4}$.

Proposition 6.8**.**

Let $m$ and $p$ be odd prime numbers with $m\leq p$ and $m\equiv p\pmod{4}$. Then there exists a $p$-sun system of $K_{mp}$.

Proof.

For each pair $(r,s)\in\mathbb{Z}^{*}_{p}\times\mathbb{Z}_{m}$, let $B_{r,s}:V(\Sigma)\rightarrow\mathbb{Z}_{p}\times\mathbb{Z}_{m}$ be the labeling of the vertices of $\Sigma$ defined as follows:

[TABLE]

Since $B_{r,s}$ is injective, for every $h\in\mathbb{Z}_{m}$ the graph S_{r,s}^{h}=\tau_{(0,h)}\big{(}B_{r,s}(\Sigma)\big{)} is a $p$-sun. For $i,j\in\mathbb{Z}_{m}$, we also notice that $\Delta_{ij}\{S_{r,s}^{h}\mid h\in\mathbb{Z}_{m}\}=\{\pm~{}r\}$ whenever $i-j=\pm s$, otherwise it is empty.

Letting $\mathcal{S}$ be the union of the following two sets of $p$-suns:

[TABLE]

it is not difficult to see that for every $i,j\in\mathbb{Z}_{m}$

[TABLE]

It is left to construct a set ${\mathcal{T}}$ of $p$-suns such that $\Delta_{ij}{\mathcal{T}}=\mathbb{Z}_{p}\setminus\Delta_{ij}\mathcal{S}$ whenever $i\neq j$, and $\Delta_{ii}{\mathcal{T}}=\mathbb{Z}^{*}_{p}\setminus\Delta_{ii}\mathcal{S}=\mathbb{Z}^{*}_{p}$. Therefore,

[TABLE]

It is enough to take ${\mathcal{T}}$ as a set consisting of one sun of type $(h,h+x)$ and $\frac{p-m}{2}$ suns of type $(h,h+1)$, for every $h\in\mathbb{Z}_{m}$ and $x\in[1,\frac{m-1}{2}]$. These $p$-suns exist by Lemma 6.4, therefore $\mathcal{S}\cup{\mathcal{T}}$ is the desired $p$-sun system of $K_{mp}$. ∎

Example 6.9**.**

Let $(m,p)=(3,11)$. Following the proof of Proposition 6.8, we construct an $11$-sun system of $K_{33}$. For every $h\in\mathbb{Z}_{3}$ and $r\in[1,3]$, let $S^{h}_{r,1}$ be the $11$-sun defined below:

[TABLE]

One can check that $\Delta_{ij}\{S^{0}_{r,1},S^{1}_{r,1},S^{2}_{r,1}\}=\{\pm r\}$ if $i\neq j$, otherwise it is empty. Therefore, letting $\mathcal{S}=\{S^{h}_{r,1}\mid h\in\mathbb{Z}_{3},r\in[1,3]\}$, we have that $\Delta_{ij}\mathcal{S}$ is non-empty only when $i\neq j$, in which case we have $\Delta_{ij}\mathcal{S}=\pm[1,3]$.

Now let ${\mathcal{T}}=\{T_{hg}\mid h\in\mathbb{Z}_{3},g\in[1,5]\}$ where $T_{hg}$ is the $11$-sun defined as follows:

[TABLE]

Note that each $T_{hg}$ is an $11$-sun of type $(h,h+1)$. Therefore we have that

[TABLE]

By Corollary 2.2, it follows that ${\mathcal{S}}\cup{\mathcal{T}}$ is an $11$-sun system of $K_{33}$.

We are now ready to show that the necessary conditions for the existence of a $p$-sun system of $K_{v}$ are also sufficient whenever $p$ is an odd prime. In other words, we end this section by proving Theorem 6.1.

Proof of Theorem 6.1. If $p=3,5$ the result can be found in [10] and in [8], respectively. For $p\geq 7$, the result follows from Propositions 6.5, 6.6 and 6.8.

Acknowledgements

The authors gratefully acknowledge support from GNSAGA of Istituto Nazionale di Alta Matematica.