On k-rainbow domination in regular graphs

TL;DR

This paper investigates the $k$-rainbow domination number in regular graphs, establishing bounds, necessary conditions for extremal cases, and exact values for specific Cayley graphs, advancing understanding of domination parameters in graph theory.

Contribution

It provides a lower bound for the $k$-rainbow domination number in $d$-regular graphs and characterizes when this bound is tight, including exact values for certain Cayley graphs.

Findings

Lower bound for $ ext{γ}_{rk}(G)$ in $d$-regular graphs

Necessary conditions for graphs to attain the bound

Exact $k$-rainbow domination numbers for cubic Cayley graphs

Abstract

The $k$-rainbow domination problem is studied for regular graphs. We prove that the $k$-rainbow domination number $\gamma_{rk}(G)$ of a $d$-regular graph for $d\leq k\leq 2d$ is bounded below by $\displaystyle{\left\lceil kn/2d\right\rceil}$, where $n$ is the order of a graph. We determine necessary conditions for regular graphs to attain this bound and find several examples. As an application, we determine exact $k$-rainbow domination numbers for all cubic Cayley graphs over abelian groups.

Tables

Table 1. Table 1: γ r ​ k ​ ( G ) subscript 𝛾 𝑟 𝑘 𝐺 \gamma_{rk}(G) -functions of G = Pr ⁡ ( m ) 𝐺 Pr 𝑚 G=\Pr(m) for k = 4 , 5 𝑘 4 5 k=4,5 , where m = 6 ​ t + s ≥ 3 𝑚 6 𝑡 𝑠 3 m=6t+s\geq 3 .

$s$	${\gamma }_{r4}\left(G\right)$		${\gamma }_{r5}\left(G\right)$
$0$	$\frac{4}{3}m$		$\frac{5}{3}m$
$1$	$\frac{4}{3}m+\frac{2}{3}$		$\frac{5}{3}m+\frac{1}{3}$
$2$	$\frac{4}{3}m+\frac{4}{3}$		$\frac{5}{3}m+\frac{2}{3}$
$3$	$\frac{4}{3}m+1$		$\frac{5}{3}m+1$
$4$	$\frac{4}{3}m+\frac{5}{3}$		$\frac{5}{3}m+\frac{4}{3}$
$5$	$\frac{4}{3}m+\frac{4}{3}$		$\frac{5}{3}m+\frac{2}{3}$

Table 2. Table 2: γ r ​ 3 ​ ( G ) subscript 𝛾 𝑟 3 𝐺 \gamma_{r3}(G) -functions for G = ML ⁡ ( m ) 𝐺 ML 𝑚 G=\operatorname{ML}(m) , where m = 6 ​ t + s ≥ 2 𝑚 6 𝑡 𝑠 2 m=6t+s\geq 2 .

$s$	${\gamma }_{r3}\left(G\right)$
$0$	$m+1$
$1$	$m+2$
$2$	$m+1$
$3$	$m$
$4$	$m+1$
$5$	$m+1$

Table 3. Table 3: γ r ​ k ​ ( G ) subscript 𝛾 𝑟 𝑘 𝐺 \gamma_{rk}(G) -functions of G = ML ⁡ ( m ) 𝐺 ML 𝑚 G=\operatorname{ML}(m) for k = 4 , 5 𝑘 4 5 k=4,5 where m = 6 ​ t + s ≥ 2 𝑚 6 𝑡 𝑠 2 m=6t+s\geq 2 .

$s$	${\gamma }_{r4}\left(G\right)$		${\gamma }_{r5}\left(G\right)$
$0$	$\frac{4}{3}m+1$		$\frac{5}{3}m+1$
$1$	$\frac{4}{3}m+\frac{5}{3}$		$\frac{5}{3}m+\frac{4}{3}$
$2$	$\frac{4}{3}m+\frac{4}{3}$		$\frac{5}{3}m+\frac{2}{3}$
$3$	$\frac{4}{3}m$		$\frac{5}{3}m$
$4$	$\frac{4}{3}m+\frac{2}{3}$		$\frac{5}{3}m+\frac{1}{3}$
$5$	$\frac{4}{3}m+\frac{4}{3}$		$\frac{5}{3}m+\frac{2}{3}$

Equations

$$\forall v\in V:f(v)=\varnothing ⟹\bigcup _{u\sim v}f(u)=\mathcal{C}.$$

$$w(f)=\sum _{v\in V}|f(v)|.$$

$${\gamma }_{rk}(G)=\min \{w(f)\mid f:V\to 2^{\mathcal{C}}\text{ is a k-RDF}\}.$$

$${\gamma }_{r{k}^{\prime }}(G)\le {\gamma }_{rk}(G)+(k^{\prime }-k)\lfloor \frac{{\gamma }_{rk}(G)}{k}\rfloor .$$

$${\gamma }_{rk}(G)\ge \lceil \frac{kn}{\Delta +k}\rceil .$$

$${\gamma }_{rk}(G)$$

$$V_i=\{v\in V:i\in f(v)\}\text{ for }i=1,\dots ,k,$$

$$C_i=\{v\in V:|f(v)|=i\}\text{ for }i=0,\dots ,k,$$

$$C=\bigcup _{i=1}^k{C}_i,$$

$$E_0$$

$$E_1$$

$$E_2$$

$$e_i=|E_i|,n_i=|V_i|,c_i=|C_i|,c=|C|=n-c_0.$$

$$e_1=cd-2e_2=c_{0}d-2e_0$$

$$c_0-\frac{2e_0}{d}$$

$$\frac{n}{2}-\frac{e_0}{d}$$

$$w(f)$$

$$w(f)\ge \frac{(k-d)}{d}n+\frac{2d-k}{d}\cdot \frac{n}{2}=\frac{kn}{2d}.$$

$$w(f)>\frac{kn/2+2e_2}{d}\ge \frac{kn}{2d}.$$

$${\gamma }_{r2}(C_n)$$

$${\gamma }_{r3}(C_n)$$

$${\gamma }_{r2}(C_n)\ge \lceil \frac{n}{2}\rceil ,{\gamma }_{r3}(C_n)\ge \lceil \frac{3n}{4}\rceil ,$$

$$c$$

$$c_0$$

$$f(u_i)=\{j\in \mathcal{C}:j\equiv i(\mathrm{mod}d)\}\text{ and }f(v_i)=\varnothing \text{ for }i=1,\dots ,d.$$

$$f^{\prime }(v)=\{\begin{array}{ll}f(v)\cup \{k+1\};& i\in f(v),\\ f(v);& i\notin f(v).\end{array}$$

$$|f(v)|\le |\{u:u\sim v,f(u)=\varnothing \}|$$

$$\lceil \frac{4m}{3}\rceil \le {\gamma }_{r4}(G)\le \lceil \frac{4m}{3}\rceil +\{\begin{array}{ll}0;& m\equiv 0,1(\mathrm{mod}6),\\ 1;& m\equiv 2,3,5(\mathrm{mod}6),\\ 2;& m\equiv 4(\mathrm{mod}6).\end{array}$$

$$4m+2=3w(f)=\sum _i{\gamma }_i\ge 4(m-2)+5\cdot 2.$$

$$\lceil \frac{5m}{3}\rceil \le {\gamma }_{r5}(G)\le \lceil \frac{5m}{3}\rceil +\{\begin{array}{ll}0;& m\equiv 0,1(\mathrm{mod}6),\\ 1;& m\equiv 2,3,4,5(\mathrm{mod}6).\end{array}$$

Full text

Keywords: Domination; Rainbow domination number; Regular graphs; Cayley graphs.

2000 Mathematics Subject Classification: 05C69, 05C85.

00footnotetext: This work is supported in part by the Slovenian Research Agency (ARRS), research program P1-0285 and research projects J1-9108, J1-1694, J1-1695.

Corresponding author e-mail: bostjan.kuzman@pef.uni-lj.si (B. Kuzman)

1 Introduction

The concept of $k$-rainbow domination, as introduced by Brešar et al. in [2], is an extension of the classical domination problem in graphs and was initiated by Hartnell and Rall [7], who studied the domination numbers of cartesian product $G\square K_{2}$ in relation to the Vizing conjecture. The problem has natural applications in analysis of networks.

Let $G=(V,E)$ be a finite, simple graph, and let $k$ be a given nonnegative integer. We denote by ${\cal C}=\{1,\ldots,k\}$ the set of colors. We say that a coloring function $f\colon V\to 2^{\cal C}$ that assigns to each vertex $v\in V$ a subset of colors $f(v)\subseteq{\cal C}$, is a $k$-rainbow dominating function (or $k$-RDF) on $G$, if

[TABLE]

In other words, every non-colored vertex $v\in V$ is $k$-rainbow dominated by its neighbours of all possible colors. For a given $k$-RDF, we define the weight of $f$ as

[TABLE]

The $k$-rainbow domination number $\gamma_{rk}(G)$ is the minimal possible weight attained by a $k$-RDF on $G$:

[TABLE]

Any $k$-rainbow domination function $f$ of weight $w(f)=\gamma_{rk}(G)$ is called a $\gamma_{rk}(G)$-function.

Clearly, any $1$-rainbow domination function on $G$ corresponds to a usual dominating set for $G$, and $1$-rainbow domination number $\gamma_{r1}(G)$ coincides with the usual domination number $\gamma(G)$ of the graph. Also, it was shown in [2], that $\gamma_{rk}(G)=\gamma(G\square K_{k})$, where $G\square K_{k}$ denotes the cartesian product of $G$ with a complete graph on $k$ vertices. For given graph $G$ and positive integer $k$, determination of the exact value $\gamma_{rk}(G)$ is known to be NP-complete [3, 4].

For $k=2,3$, the exact values and upper bounds for $k$-rainbow domination numbers of specific graph families such as the generalized Petersen graphs, trees, products of paths and cycles, grid graphs, etc, were studied in several papers, see for instance [3, 4, 8, 9, 11, 12]. Moreover, general upper bounds for $k$-rainbow domination numbers of connected graphs are also known and were proven to be tight for some specific graphs, see [5, 6].

However, there are few results available on determining the full set of $k$-rainbow dominaton numbers of particular graph families for all relevant $k$. In order to determine these for some graph $G$, one could first determine $\gamma_{r1}(G)$ and then apply the following theorem to obtain some lower and upper bounds for $\gamma_{rk}(G)$ for $k\geq 2$.

Theorem 1.1** (Shao et al., 2014).**

Let $G=(V,E)$ be a connected graph of order $n$.

1. (i)

If $k^{\prime}>k$, then

[TABLE] 2. (ii)

If $\Delta$ denotes the maximal degree in graph $G$, then

[TABLE]

In this paper, we focus on $k$-rainbow domination numbers of regular graphs. In Section 2, we prove the following Theorem, which improves the lower bound of Theorem 1.1 for $d$-regular graphs, whenever $d\leq k\leq 2d$.

Theorem 1.2**.**

Let $G$ be a $d$-regular graph of order $n$. Then

[TABLE]

and $\gamma_{rk}(G)=n\text{ for }k\geq 2d$.

In Section 2, we also prove several other inequalities and interesting results for $k$-rainbow domination functions of regular graphs. In Section 3, we define a $d$-rainbow domination regular graph as a $d$-regular graph such that $\gamma_{rk}(G)$ attains the lower bounds from Theorem 1.2 for all $d\leq k\leq 2d$ and investigate necessary conditions for parameters of such graphs. In Section 4, we investigate exact $k$-rainbow domination numbers of all connected cubic Cayley graphs over abelian groups for all $k$ and determine all $3$-rainbow domination regular graphs among these. In Section 5, some further examples of $4$-rainbow domination regular graphs are given and some open questions are posed.

2 Lower bounds for regular graphs

Unless otherwise noted, throughout this section, graph $G=(V,E)$ will be regular of order $n$ and degree $d$. By using elementary counting arguments, we shall obtain certain bounds on the weight of any $k$-rainbow domination function $f$ on $G$.

Suppose that graph $G=(V,E)$ and some function $f\colon V\to 2^{\cal C}$ are given. In what follows, we denote the non-disjoint sets of vertices that are colored with color $i\in{\cal C}$ by

[TABLE]

and by $V_{0}=\{v\in V\colon f(v)=\emptyset\}$ the set of non-colored vertices. Further, we denote the disjoint sets of vertices which are colored with exactly $i$ different colors by

[TABLE]

their union of all colored vertices by

[TABLE]

and the sets of edges with exactly $i$ end-vertices colored by

[TABLE]

Also, we denote the sizes of respective sets by

[TABLE]

Our first observation is obtained by double counting the elements of $E_{1}$ in two different ways, and then omitting some terms to obtain upper and lower bounds for the number of $f$-colored vertices $c$.

Lemma 2.1**.**

Let $G$ be a $d$-regular graph of order $n$ and let $f$ be some $k$-rainbow domination function on $G$. Then

[TABLE]

and the following inequalities hold:

[TABLE]

Next, we use another double counting argument to get several lower bounds for the weight of a $k$-RDF.

Lemma 2.2**.**

Let $G$ be a $d$-regular graph of order $n$ and let $f$ be some $k$-rainbow domination function on $G$. Then the following inequalities hold:

[TABLE]

Proof.

First, we count the number of ordered triples $(u,v,i)$, where $u\sim v$, $f(u)=\emptyset$ and $i\in f(v)$. Since each empty vertex is dominated by at least $k$ neighbours of different colors, we see this number is at least $(n-c)k$. On the other hand, there are exactly $w(f)$ pairs $(v,i)$ with $i\in f(v)$, and each of these has $d$ neighbours, some colored, others non-colored. Subtracting twice the number of edges with both edges colored we get the exact number of triples. Therefore, $(n-c)k\leq w(f)d-2e_{2}$, and inequality (5) follows. Since $w(f)\geq c$, inequality (6) is easily obtained from (5). Eliminating $2e_{2}$ from inequalities (5) and (3), we combine them to obtain (7) and then rewrite with $c=n-c_{0}$ to obtain (8). ∎

We merge Lemmas 2.1 and 2.2 to obtain the following Proposition.

Proposition 2.3**.**

Let $G$ be a $d$-regular graph of order $n$ and let $f$ be some $k$-rainbow domination function on $G$. Then

[TABLE]

and $w(f)\geq n$ for $k\geq 2d.$

Proof.

For $k<2d$, we check two cases. If $c\geq n/2$, we apply inequality (7) of Lemma 2.2 to get

[TABLE]

If $c<n/2$, then $n-c>n/2$ and we apply Lemma 2.1 to get

[TABLE]

In both cases, $w(f)$ is an integer, greater or equal to $\frac{kn}{2d}$.

For $k=2d$, we get $w(f)\geq n$ directly from (7). For $k>2d$, we get $w(f)\geq 2(n-c)+\frac{2e_{2}}{d}$ from (5) and hence $w(f)\geq n$ from (3). ∎

Since for any $\gamma_{rk}(G)$-function $f$ we have $w(f)=\gamma_{rk}(G)\leq n$, Theorem 1.2 is a direct corollary of Proposition 2.3 and needs no further proof.

Example 2.4**.**

Exact $k$-rainbow domination numbers of cycles $C_{n}$, $n\geq 3$, were determined for $k=2$ by Brešar and Kraner [3] and for $k=3$ by Shao et al. [9] as

[TABLE]

Using Theorem 1.2, we get $\gamma_{rk}(C_{f}n)=n$ for all $k\geq 4$ and lower bounds

[TABLE]

*which can be used to shorten the original proofs significantly. *

We can also rewrite inequality (7) to obtain bounds for parameters $c$ and $c_{0}$.

Proposition 2.5**.**

Let $G$ be a $d$-regular graph and let $f$ be a $\gamma_{rk}(G)$-function, where $0<k<2d$. Then

[TABLE]

3 $d$-rainbow domination regular graphs

The next theorem gives necessary conditions for a $d$-regular graph to attain the lower bound for $\gamma_{rk}(G)$.

Theorem 3.1**.**

Let $G$ be a $d$-regular graph of order $n$ and let $k<2d$. If $\gamma_{rk}(G)=\frac{kn}{2d}$, then $k\geq d$, $2d|n$ and $G$ is bipartite.

Proof.

For $k<2d$, let $w(f)=\frac{kn}{2d}$ for some $\gamma_{rk}(G)$-function $f$. Then $c<n/2$ is not possible by (5) and $c\geq n/2$ forces $c=n/2$ by (7). Since $\gamma_{rk}(G)\geq c$, we also get $k\geq d$. Moreover, it follows from (5) and (2) that $e_{0}=e_{2}=0$ and $e_{1}=e$, so $G$ is bipartite with bipartition sets $C_{0},C$ of size $n/2$. Now, count pairs $(i,u)$, such that $i\in{\cal C}$, $u\in V$, $f(u)=\emptyset$. Their number equals $v_{i}d$ and is at least $c_{0}=n/2$, so $v_{i}\geq n/2d$. But $w(f)=\sum iv_{i}\geq kn/2d$ implies equality $v_{i}=n/2d$ for all $i>0$, so $2d|n$. (Observe that the proof also implies that for any $\gamma_{rk}(G)$-function $f$, equality $v_{i}=n/2d$ must hold for all colors $i\in{\cal C}$.) ∎

Example 3.2**.**

The Franklin graph $F$ in Figure 1 shows that conditions of Theorem 3.1 are not sufficient. Since assigning a color $i\in{\cal C}$ to any two vertices in one bipartition set cannot dominate all 6 vertices in the other bipartition set, we get $\gamma_{r3}(F)>6$.

Example 3.3**.**

On the other hand, it is easy to see that the complete bipartite graph $G=K_{d,d}$ satisfies conditions of Theorem 3.1 and the equality $\gamma_{rk}(G)=\frac{kn}{2d}$ holds for all $d\leq k\leq 2d$. Indeed, for $d\leq k\leq 2d$, denote by $u_{i},v_{i}\in V(G)$ the vertices in the bipartition sets and let

[TABLE]

Then each color appears exactly once and dominates $d$ non-colored vertices, so $f$ is a $k$-RDF of weight $w(f)=k=k\frac{n}{2d}$.

The ‘rainbow property’ of graphs $K_{d,d}$ described above can be generalized as follows. We define that graph $G$ is $d$-rainbow domination regular (or $d$-RDR), if it is $d$-regular and equality $\gamma_{rd}(G)=\frac{|V(G)|}{2}$ holds. We now show this also implies a stronger condition.

Proposition 3.4**.**

Suppose that $G$ is a $d$-regular graph of order $n$, and let $d\leq k<2d$ for some $k$. If $\gamma_{rk}(G)=\frac{kn}{2d}$, then $\gamma_{r(k+1)}(G)=\frac{(k+1)n}{2d}$.

Proof.

Suppose that $d\leq k<2d$ and $f\colon V\to\{1,\ldots,k\}$ is $\gamma_{rk}(G)$-function with $\gamma_{rk}(G)=\frac{kn}{2d}$. Then $v_{i}=\frac{n}{2d}$ for all $i=1,\ldots,k$ by proof of Theorem 3.1 and a $\gamma_{r(k+1)}$-function $f^{\prime}$ can be explicitly constructed from $f$ by selecting (any) color $i\in\{1,\ldots,k\}$ and adding color $k+1$ to all $i$-colored vertices, that is, defining

[TABLE]

Obviously, $f^{\prime}$ is a $(k+1)$-RDF of weight $w(f^{\prime})=\sum_{i}v_{i}=\frac{(k+1)n}{2d}$. ∎

Corollary 3.5**.**

Let $G$ be a $d$-regular graph of order $n$. Then $G$ is $d$-rainbow domination regular if and only if $\gamma_{rk}(G)=\frac{kn}{2d}$ for all $d\leq k\leq 2d$.

Example 3.6**.**

For any $d\geq 1$, complete bipartite graph $K_{d,d}$ is a $d$-RDR graph. It is not difficult to see that a cycle $C_{n}$ is a $2$-RDR graph if and only if $n\equiv 0\pmod{4}$.

We shall identify some more examples of $d$-RDR graphs in the next section.

4 Cubic Cayley graphs over abelian groups

In this section, we study the $k$-rainbow domination numbers for connected cubic (that is, $3$-regular) Cayley graphs over some finite abelian group $H$. Our motivation comes from the fact that Cayley graphs are a large class of regular graphs with nice symmetry properties that could provide some more examples of $d$-RDR graphs, and also from some existing results on values and bounds for $2$- and $3$-rainbow domination numbers of generalised Petersen graphs that partially overlap with our class and could be extended to larger $k$. For cubic graphs, we readily have $\gamma_{rk}(G)\geq\frac{kn}{6}$ for $3\geq k<6$ and $\gamma_{rk}(G)=n$ for $k\geq 6$ by Theorem 1.2.

Recall that for any finite group $H$ and inverse closed subset $S=S^{-1}\subseteq H\setminus\{1_{H}\}$, the (non-directed and simple) Cayley graph $G=\operatorname{Cay}(H,S)$ is defined by vertex set $V(G)=H$ and edge set $E(G)=\{\{g,h\}\colon g,h\in H,gh^{-1}\in S\}$. Graph $G=\operatorname{Cay}(H,S)$ is regular of degree $d=|S|$ and $G$ is connected if and only if $S$ is a generating set for group $H$.

Note that in order for $\operatorname{Cay}(H,S)$ to be cubic, we must have either $S=\{a,a^{-1},b\}$, where $a\neq a^{-1}$ and $b=b^{-1}$, or $S=\{a,b,c\}$, where $a=a^{-1}$, $b=b^{-1}$ and $c=c^{-1}$. This implies the following well-known classification of such graphs. We omit details of the proof.

Proposition 4.1**.**

Let $G=\operatorname{Cay}(H,S)$ be a connected cubic Cayley graph over some finite abelian group $H$. Then $G$ is isomorphic either to a $m$-sided prism $\Pr(m)$ or to a Möbius ladder $\operatorname{ML}(m)$, where $m=\frac{|H|}{2}\geq 2$ and graphs $\Pr(m)$ and $\operatorname{ML}(m)$ of order $n=2m$ both have vertex set $V=\{u_{i},v_{i}\colon i\in\mathbb{Z}_{m}\}$, while the edge set is equal to $E=\{\{u_{i},v_{i}\},\{u_{i},u_{i+1}\},\{v_{i},v_{i+1}\}\colon i\in\mathbb{Z}_{m}\}$ for prisms $\Pr(m)$ and to $E=\{\{u_{i},v_{i}\},\{u_{i},u_{i+1}\},\{v_{i},v_{i+1}\}\colon i\in\mathbb{Z}_{m},i\neq m-1\}\cup\{\{u_{m-1},v_{m-1}\},\{u_{m-1},v_{0}\},\{v_{m-1},u_{0}\}\}$ for Möbius ladders $\operatorname{ML}(m)$.

This reduces the problem to determination of $\gamma_{rk}(G)$ for prisms and Möbius ladders. We shall state and prove separate theorems for each case. Note that the prisms $\Pr(m)$ are just generalised Petersen graphs $GP(m,1)$, so for $k=2,3$, their $k$-rainbow domination numbers were already determined in [9, 10].

Theorem 4.2**.**

Let $G=\Pr(m)$ be the $m$-sided prism of order $n=2m$, $m\geq 3$. Then

• •

$\gamma_{r1}(G)=\left\lceil\frac{m}{2}\right\rceil+\begin{cases}1;&m\equiv 2\pmod{4},\\ 0;&m\equiv 0,1,3\pmod{4}.\end{cases}$**

• •

$\gamma_{r2}(G)=m$.

• •

$\gamma_{r3}(G)=m+\begin{cases}0;&m\equiv 0\pmod{6},\\ 1,&m\equiv 1,2,3,5\pmod{6},\\ 2;&m\equiv 4\pmod{6}.\\ \end{cases}$**

• •

$\gamma_{r4}(G)=\left\lceil\frac{4m}{3}\right\rceil+\begin{cases}0;&m\equiv 0,1\pmod{6},\\ 1;&m\equiv 2,3,4,5\pmod{6},\\ \end{cases}$**

• •

$\gamma_{r5}(G)=\left\lceil\frac{5m}{3}\right\rceil+\begin{cases}0;&m\equiv 0,1,2,5\pmod{6},\\ 1;&m\equiv 3,4\pmod{6}.\\ \end{cases}$**

• •

$\gamma_{rk}(G)=2m$* for $k\geq 6.$*

Moreover, for $m\equiv 0\pmod{6}$ and $3\leq k\leq 6$, the lower bound $\gamma_{rk}(G)=\frac{km}{3}$ is attained. For $k=4,5$, appropriate $\gamma_{rk}(G)$-functions are given in Table 1.

Before providing the proof, we state the following observation, which is valid for any graph $G$ (also non-regular) and is sometimes essential for finding exact values of $\gamma_{rk}(G)$.

Lemma 4.3** (Discharging principle for $k$-rainbow domination).**

If a $\gamma_{rk}(G)$-function $f$ on $G$ has minimal number of non-colored vertices (that is, size $|\{v\colon f(v)=\emptyset\}|$ is minimal among all $\gamma_{rk}(G)$-functions on $G$), then

[TABLE]

for all $v\in V(G)$.

Proof.

If the condition is not true for some $v\in V$, reassigning (”discharging”) some colors from $v$ to its non-colored neighbours reduces $c_{0}$ without changing $w(f)$. ∎

Proof of Theorem 4.2.

For $k=1$, the result is well-known [1] and for $k=2,3$, the results are proven in [9, 10]. The case $k\geq 6$ is also clear.

Let $k=4$. Using Theorem 1.2 for the lower bound and Theorem 1.1 with $t=3=t^{\prime}+1$ for the upper bound, we obtain

[TABLE]

For $m=0,1\pmod{6}$, we obviously have $\gamma_{r4}(G)=\left\lceil\frac{4n}{3}\right\rceil$. For $m=3\pmod{6}$, the prism $G=\Pr(m)$ is not bipartite, so we have a strict inequality $\left\lceil\frac{4m}{3}\right\rceil=\frac{4m}{3}<\gamma_{r4}(G)$ by Theorem 3.1 , and $\gamma_{r4}(G)=\left\lceil\frac{4n}{3}\right\rceil+1$ follows. It is not difficult to find appropriate $\gamma_{r4}(G)$-functions in these 3 cases, see Table 1.

For $m\equiv 2,5\pmod{6}$, suppose that $\gamma_{r4}(G)=\left\lceil\frac{4n}{3}\right\rceil$, and let $f$ be some $\gamma_{r4}(G)$-function. Using (8) we obtain $c_{0}\geq m-1/2$, hence $c_{0}\geq m$. Since $m=6t+2$ or $6t+5$ and every color on a vertex dominates at most 3 empty vertices, this implies that $n_{i}\geq 2t+1$ or $n_{i}\geq 2t+2$, respectively, for $i=1,2,3,4$. But then $w(f)\geq 8t+4$ or $8t+8$, which is larger than $\gamma_{r4}(G)$, a contradiction. Hence $\gamma_{r4}(G)=\left\lceil\frac{4n}{3}\right\rceil+1$ and appropriate $\gamma_{r4}(G)$-functions are easily constructed from the known $\gamma_{r3}(G)$-functions, see Table 1 and Remark 4.4.

Finally, for $m\equiv 4\pmod{6}$, we have $\gamma_{rk}(G)=\left\lceil\frac{4m}{3}\right\rceil+a$ with $a=0,1$ or $2$. Suppose first that $a=0$ and let $f$ be some $\gamma_{rk}(G)$-function. Then by (8), we get $c_{0}\geq m-1$. If $c_{0}\geq m=6t+4$, we have $n_{i}\geq 2t+2$ for all $i$, so $w(f)\geq 8t+8>8t+6=\left\lceil 4m/3\right\rceil$. So $a=0$ implies $c_{0}=m-1$ and $c=m+1$. From (5) we further obtain $e_{2}=3$ and $e_{0}=0$, so every edge has at least one colored vertex. This implies we may wlog (without loss of generality) suppose that the set of noncolored vertices is $C_{0}=\{u_{0},u_{2},\ldots,u_{m-2},v_{1},v_{3},\ldots,v_{m-3}\}$ (note that $f(u_{m-1}),f(v_{m-1})\notin C_{0}$). As in the proof of Theorem 4 in [9], we now consider subsets $\mathcal{P}_{i}=\{v_{i},v_{i+1},v_{i-1},u_{i},u_{i+1},u_{i-1}\}$ for $i\in\mathbb{Z}_{m}$, and denote by $\gamma_{i}=\sum_{v\in\mathcal{P}_{i}}|f(v)|$ the $f$-weight of $\mathcal{P}_{i}$. Observe that $\gamma_{0}\geq 5$, since $u_{0}$ is dominated by $4$ colors and $v_{m-1}\neq\emptyset$. Similarily, $\gamma_{m-2}\geq 5$, while $\gamma_{i}\geq 4$ for all other $i$. Since each vertex belongs to exactly three blocks $\mathcal{P}_{i}$, we see that

[TABLE]

Hence, equality holds and we have $\gamma_{i}=4$ for $i\neq 0,m-2$. In particular, $\gamma_{m-1}=4$ implies $|f(u_{m-1})|\leq 1$ and $|f(v_{0})|\leq 1$, and $\gamma_{0}=5$ further implies $|f(u_{1})|=2$. Continuing, we see that $|f(v_{4})|=|f(u_{7})|=\ldots=|f(u_{m-3})|=2.$, while $|f(v)|=1$ for all other vertices from $C$. But now, it is easy to see that any choice of colors forces $f(u_{m-1})=f(v_{m-2})$, hence $u_{m-2}$ is not dominated by all colors. This contradiction shows $a\geq 1$, but in fact, $a=1$, as we can construct an appropriate $4$-RD function of weight $\left\lceil\frac{4m}{3}\right\rceil+1$, see Table 1.

Now, let $k=5$. Using the known bounds and values for $\gamma_{r4}(G)$, with some computation we obtain

[TABLE]

As in case $k=4$, the values of $\gamma_{rk}(G)$ are obtained trivially for $m\equiv 0,1,3\pmod{6}$.

For $m\equiv 2,4,5\pmod{6}$, write $m=6t+2,4,5$, resp., and suppose that $\gamma_{rk}(G)=\left\lceil 5n/3\right\rceil=10t+4,7,9$, respectively. Let $f$ be a $\gamma_{rk}(G)$-function. It follows from (8) that $c_{0}\geq m-a$, with $a=2,1,2$ for $m\equiv 2,4,5\pmod{6}$, respectively. If $c_{0}\geq m$, we get $n_{i}\geq 2t+1,2,2$, resp., hence $w(f)\geq 5n_{i}=10t+5,10,10$, a contradiction. Similarly, we get that $c=m-1$ is not possible for $m=2,5\pmod{6}$, while for $m=4\pmod{6}$ we apply (5) to get $e_{2}\leq 3$. Moreover, it now follows from (2) that $e_{2}-e_{0}=3$, implying that $e_{2}=3$ and $e_{0}=0$. As in case $k=4$, we now wlog suppose that $C_{0}=\{u_{0},u_{2},\ldots,u_{m-2},v_{1},v_{3},\ldots,v_{m-1}\}$ and observe sets $\mathcal{P}_{i}$ with weights $\gamma_{i}$ to get $\gamma_{0},\gamma_{m-2}\geq 6$ and $\gamma_{i}\geq 4$ otherwise. From $3w(f)=\sum_{i}\gamma_{i}$ we get that the equalities hold, implying that $|f(u_{1})|=|f(v_{4})|=\ldots=|f(u_{m-3})|=3$ and $|f(v)|=1$ for other $v\notin C_{0}$, implying further that $f(v_{0})=f(u_{m-1})$, a contradiction. Finally, suppose $c=m-2$ with $m=6t+2$ or $5$. Repeating above arguments, we get that this case is possible with $e_{2}=6$, $e_{0}=0$. Inspecting possible sets $\mathcal{P}_{i}$ and weights $\gamma_{i}$, it is now not difficult to construct appropriate $\gamma_{rk}(G)$-functions for both cases. ∎

Remark 4.4**.**

It follows from the proof of Theorem 1 in [9], that whenever equality $\gamma_{r(k+1)}(G)=\gamma_{rk}(G)+\left\lfloor\gamma_{rk}(G)/k\right\rfloor$ holds and a $\gamma_{rk}(G)$-function is known, a $\gamma_{r(k+1)}$-function $f^{\prime}$ can be explicitly constructed from $f$ by selecting color $i\in{\cal C}=\{1,\ldots,k\}$ such that $n_{i}=\min\{n_{1},\ldots,n_{k}\}$ and adding color $k+1$ to all $i$-colored vertices, that is, defining

[TABLE]

For Möbius ladders, we obtain very similar results.

Theorem 4.5**.**

Let $G=\operatorname{ML}(m)$ be the Möbius ladder of order $n=2m$, $m\geq 2$. Then

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$\gamma_{r1}(G)=\left\lceil\frac{m}{2}\right\rceil+\begin{cases}1;&m\equiv 0\pmod{4},\\ 0;&m\equiv 1,2,3\pmod{4}.\end{cases}$**

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$\gamma_{r2}(G)=m$.

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$\gamma_{r3}(G)=m+\begin{cases}0;&m\equiv 3\pmod{6},\\ 1,&m\equiv 0,2,4,5\pmod{6},\\ 2;&m\equiv 1\pmod{6}.\\ \end{cases}$**

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$\gamma_{r4}(G)=\left\lceil\frac{4m}{3}\right\rceil+\begin{cases}0;&m\equiv 3,4\pmod{6},\\ 1;&m\equiv 0,1,2,5\pmod{6},\\ \end{cases}$**

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$\gamma_{r5}(G)=\left\lceil\frac{5m}{3}\right\rceil+\begin{cases}0;&m\equiv 2,3,4,5\pmod{6},\\ 1;&m\equiv 0,1\pmod{6}.\\ \end{cases}$**

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$\gamma_{rk}(G)=2m$* for $k\geq 6.$*

Moreover, for $m\equiv 3\pmod{6}$ and $3\leq k\leq 6$, the lower bound $\gamma_{rk}(G)=\frac{km}{3}$ is attained. For $k=3,4,5$, appropriate $\gamma_{rk}(G)$-functions are given in Table 3.

Proof.

For $k=1$, the result is well known, see [1]. For $k=2$, we can define a $2$-RDF $f$ of weight $w(f)=m$ on $G$ by setting $f(u_{2i})=\{1\}$, $f(v_{2i+1})=\{2\}$ and $f(v)=\emptyset$ otherwise, so $\gamma_{r2}(G)\leq m$. Now suppose $\gamma_{r2}(G)\leq m-1$ and let $f$ be an appropriate $\gamma_{r2}(G)$-function of weight $w(f)\leq m-1$. Note that we can wlog suppose that $f$ has the minimal number of non-colored vertices and apply Lemma 4.3 (Discharging principle) whenever needed. Denote by $\mathcal{P}_{i}=\{u_{i},v_{i},u_{i+1},v_{i+1},u_{i-1},v_{i-1}\}$ and $\gamma_{i}=\sum_{v\in\mathcal{P}_{i}}|f(v)|$ for $i\in\mathbb{Z}_{m}$. Observe first that $\gamma_{i}\geq 0,1$ is not possible, so $\gamma_{i}\geq 2$. Now denote by $\alpha$ the number of blocks $\mathcal{P}_{i}$ with $\gamma_{i}=2$ and by $\beta$ the number of blocks with $\gamma_{i}\geq 4$. Then we have

[TABLE]

hence $\alpha\geq 3+\beta$. Now let $\gamma_{i}=2$ and inspect the possible values of $f$ on $\mathcal{P}_{i}$.

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It is easy to see that blocks with values $\left(\begin{smallmatrix}f(u_{i-1})&f(u_{i})&f(u_{i+1})\\ f(v_{i-1})&f(v_{i})&f(v_{i+1})\end{smallmatrix}\right)$ equal to $\left(\begin{smallmatrix}12&0&0\\ 0&0&0\end{smallmatrix}\right),$ $\left(\begin{smallmatrix}1&0&0\\ 2&0&0\end{smallmatrix}\right),$ $\left(\begin{smallmatrix}1&0&0\\ 1&0&0\end{smallmatrix}\right),$ $\left(\begin{smallmatrix}1&0&0\\ 0&0&2\end{smallmatrix}\right),$ $\left(\begin{smallmatrix}1&0&0\\ 0&0&1\end{smallmatrix}\right)$, $\left(\begin{smallmatrix}1&0&0\\ 0&1&0\end{smallmatrix}\right)$ or reflections of such blocks are not possible.

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The blocks of types $\left(\begin{smallmatrix}0&12&0\\ 0&0&0\end{smallmatrix}\right),$ $\left(\begin{smallmatrix}0&1&0\\ 0&2&0\end{smallmatrix}\right),$ $\left(\begin{smallmatrix}0&1&0\\ 0&1&0\end{smallmatrix}\right),$ are possible, but for each of them we have $\gamma_{i\pm 1}\geq 4$, so each their occurence also increases $\beta$. Since $\alpha\geq\beta+3$, we may wlog assume that there are no such blocks.

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Thus, we have at least $\alpha$ blocks of type $\left(\begin{smallmatrix}0&1&0\\ 2&0&0\end{smallmatrix}\right)$ or $\left(\begin{smallmatrix}0&1&0\\ 0&0&2\end{smallmatrix}\right).$ However, for each such block $\mathcal{P}_{i}$ either $\gamma_{i+1}$ or $\gamma_{i-1}$ is at least $4$. If $\gamma_{i}=\gamma_{i+1}=2$, then $\gamma_{i-1}=4$ and $\gamma_{i+2}=4$, hence the average weight of these four blocks is $3$. If $\gamma_{i}=2$ and $\gamma_{i+1}=3$, then we must have $\gamma_{i-1}=4$ and $\gamma_{i-2}=3$, and their average weight is again $3$. Hence, removing any such quadruplet of blocks from the equation still forces $\alpha\geq\beta+3$, but now there are no more possibilities for $\mathcal{P}_{i}$ with $\gamma_{i}=2$, a contradiction.

Let $k=3$. Then $\left\lceil nk/2d\right\rceil=m$ is the lower bound for $\gamma_{r3}(G)$ for all $m$. However, for the lower bound to be exact we must have $6|2m$ and $G$ bipartite, which only happens for $m=3\pmod{6}$. For $m=0,1,4,5\pmod{6}$, we have $\gamma_{r3}(G)\geq m+1$, but in fact equality holds, as appropriate functions are easily constructed, see Table 2.

It remains to show that $\gamma_{r3}=m+2$ for $m=6t+1$. Again, by Table 2 we confirm that this is the upper bound. Now suppose $w(f)=m+1=6t+2$ for some $3$-RDF $f$. Then $c\leq w(f)$ and hence $c_{0}\geq 6t$. Since each colored vertex dominates at most $3$ non-colored vertices, we must have $n_{1},n_{2},n_{3}\geq 2t$, but also $n_{1}+n_{2}+n_{3}=6t+2$, so $n_{i}=2t$ for at least one $i$. Wlog assume $n_{3}=2t$. Then $c_{0}\leq 6t$ and hence $c_{0}=6t$ and $c=6t+2=w(f)$, which implies $c_{1}=c$ and $c_{2}=c_{3}=0$, so each vertex is colored with at most $1$ color. By checking all possible cases for adjacent pairs of vertices, that is, $u_{i}\sim v_{i}$, $u_{i}\sim u_{i+1}$, or $u_{m-1}\sim v_{0}$, it is now easy to see that $f(u)=f(v)=\emptyset$ for some pair $u\sim v$ implies $|f(w)|=2$ for some $w$ adjacent to $u$ or $v$, a contradiction. Thus, we have $e_{0}=0$ and therefore $e_{2}=3$, so we have exactly 3 edges with both end-vertices colored (and hence no 4-cycle with all vertices colored). Note also, that since $n_{3}=2t$, color $3$ cannot be used on such edges.

We now check all different pairs of $u\sim v$ with $|f(u)|=|f(v)|=1$ to arrive at the contradiction. (Alternatively, we could denote $\mathcal{P}_{i}=\{u_{i},v_{i},u_{i+1},v_{i+1},u_{i-1},v_{i-1}\}$ and $\gamma_{i}=\sum_{v\in\mathcal{P}_{i}}|f(v)|$ to see that $3\leq\gamma_{i}\leq 4$ and $\sum_{i}\gamma_{i}=3w(f)$ implies that $\gamma_{i}=4$ for exactly $3$ consecutive $i$, and get the contradiction after some further inspection.) First, we check pairs $u_{i}\sim v_{i}$ with $i=1,\ldots,m-2$. Wlog $f(u_{1})=f(v_{1})\neq 0$. Then exactly one of $u_{0},v_{0}$ is non-colored, say $f(u_{0})=\emptyset,f(v_{0})\neq\emptyset$. Also, exactly one of $u_{2},v_{2}$ is non-colored:

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If $f(u_{2})=\emptyset$, we have $f(v_{2})\neq\emptyset$ and so $v_{0}v_{1},v_{1}u_{1},v_{2}v_{3}$ are the 3 edges with both end-vertices colored, therefore exactly one of $u_{i},v_{i}$ is noncolored for all other $i$. This implies $f(v_{3})=f(u_{4})=f(v_{5})=\ldots=f(v_{n-2})=f(u_{n-1})=\emptyset$. Since $f(v)=\{3\}$ only for vertices with 3 non-colored neighbours, we have $\{3\}=f(u_{3})=f(v_{6})=\ldots=f(v_{m-1})$. Now take wlog $f(v_{2})=\{2\}$ and $f(u_{1})=\{1\}$ and see this forces $\{2\}=f(v_{2})=f(u_{5})=\ldots=f(u_{1})$, a contradiction.

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If $f(u_{2})\neq\emptyset$, we have $f(v_{2})=\emptyset$ and so $v_{0}v_{1},v_{1}u_{1},u_{1}u_{2}$ are the 3 edges with both end-vertices colored. Again exactly one of $u_{i},v_{i}$ is noncolored for all other $i$ implying that $f(v_{m-1})=\emptyset$, a contradiction, since $v_{m-1}\sim u_{0}$ and $f(u_{0})=\emptyset$. so for all other $j$, we have exactly one of $u_{j},v_{j}$ colored.

In similar fashion, we obtain a contradiction in all other cases. Therefore $\gamma_{r3}(G)=m+2$ for $m\equiv 1\pmod{6}$.

For $k=4,5$, the proof is similar as the proof for prisms. For instance, we use Theorems 1.1 and 1.2 to obtain

[TABLE]

Then we find an appropriate $4$-RDF (see Table 3) and apply different combinatorial arguments to prove that lower weight is not possible. We omit further details. ∎

Corollary 4.6**.**

The only connected $3$-rainbow domination regular Cayley graphs over abelian groups are prisms $\Pr(m)$, where $m\equiv 0\pmod{6}$, and Möbius ladders $\operatorname{ML}(m)$, where $m\equiv 3\pmod{6}$.

5 Concluding remarks

The investigations of $k$-rainbow domination numbers for $d$-regular graphs from previous sections can be naturally generalized to some other classes of cubic graphs or to specific classes of $d$-regular graphs with $d\geq 4$. However, determining exact $k$-rainbow domination numbers for all $4$-regular Cayley graphs over abelian groups might turn out to be quite difficult already.

It would also be interesting to obtain further classification of $d$-RDR graphs. For two more examples, see Figure 2. It is easy to check that the tesseract graph $Q_{4}$ is a $4$-RDR graph, and the wreath graph $G=C_{m}[2K_{1}]$ (the lexicographic product of a cycle with 2 isolated vertices) is a $4$-RDR graph for all $m\geq 3$ with $m\equiv 0\pmod{4}$.

We observe that in both cases, the graphs are Cayley graphs over abelian groups, namely $Q_{4}\cong\operatorname{Cay}(\mathbb{Z}_{4},\{e_{1},e_{2},e_{3},e_{4}\})$ and $C_{m}[2K_{1}]\cong\operatorname{Cay}(\mathbb{Z}_{m}\times\mathbb{Z}_{2},\{(\pm 1,0),(\pm 1,1)\})$. We can thus ask the following questions:

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Question 1: Are there any $d$-RDR graphs that are not obtained as Cayley graphs over some abelian group?

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Question 2: More generaly, are there any $d$-RDR graphs that are not vertex transitive?

6 Acknowledgments

This work is supported in part by the Slovenian Research Agency (ARRS), research program P1-0285 and research projects J1-9108, J1-1694, J1-1695. The author also thanks to Primož Šparl for suggesting the problem.