On degree sum conditions for 2-factors with a prescribed number of cycles

TL;DR

This paper establishes new degree sum conditions in highly connected graphs that guarantee the existence of a 2-factor with a specified number of cycles, generalizing previous results in the field.

Contribution

It introduces a generalized degree condition involving maximum degree sums of independent sets that ensures the existence of 2-factors with a given number of cycles, extending prior work.

Findings

Proves a degree sum condition for 2-factors with exactly k cycles

Generalizes previous results by Brandt et al. and Yamashita

Applicable to m-connected graphs of order n ≥ 5k - 2

Abstract

For a vertex subset $X$ of a graph $G$, let $\Delta_{t}(X)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k$ be a positive integer, and let $G$ be an $m$-connected graph of order $n \ge 5k - 2$. If $\Delta_{2}(X) \ge n$ for every independent set $X$ of size $\lceil m/k \rceil+1$ in $G$, then $G$ has a 2-factor with exactly $k$ cycles. This is a common generalization of the results obtained by Brandt et al. [Degree conditions for 2-factors, J. Graph Theory 24 (1997) 165-173] and Yamashita [On degree sum conditions for long cycles and cycles through specified vertices, Discrete Math. 308 (2008) 6584-6587], respectively.

Tables

Table 1. Table 1: Comparison of the degree conditions

	hamilton cycle	2-factor with $k$ cycles
Ore-type	${\sigma }_2\ge n$	${\sigma }_2\ge n$
	Theorem A (Ore)	Theorem E (Brandt et al.)
Chvátal-Erdős-type	$\alpha \le \kappa$	$\alpha \le \lceil \kappa /k\rceil$
	Theorem B (Chvátal and Erdős)	Remark 2
Bondy-type	${\sigma }_{\kappa +1}>\frac{1}{2}\left(\kappa +1\right)\left(n-1\right)$	${\sigma }_{\lceil \kappa /k\rceil +1}>\frac{1}{2}\left(\lceil \kappa /k\rceil +1\right)\left(n-1\right)$
	Theorem C (Bondy)	Remark 2
Yamashita-type	${\sigma }_2^{\kappa +1}\ge n$	${\sigma }_2^{\lceil \kappa /k\rceil +1}\ge n$
	Theorem D (Yamashita)	Theorem 1 (Main theorem)

Equations

$$\displaystyle\sigma_{m}(G)=\min\Big{\{}\sum_{x\in X}d_{G}(x):\textup{$X$ is an independent set of $G$ with $|X|=m$}\Big{\}};$$

$$\displaystyle\Delta_{t}(X)=\max\Big{\{}\sum_{x\in Y}d_{G}(x):Y\subseteq X,\ |Y|=t\Big{\}}.$$

$$\displaystyle\sigma_{t}^{m}(G)=\min\Big{\{}\Delta_{t}(X):\textup{$X$ is an independent set of $G$ with $|X|=m$}\Big{\}};$$

$$\displaystyle|C^{*}|\ \big{(}=\sum_{1\leq p\leq k}|C_{i}|\big{)}\textup{ is as large as possible.}$$

$$u_1,u_2,\dots ,u_l$$

$$\displaystyle\overrightarrow{P}=\left\{\begin{array}[]{ll}x\overrightarrow{C_{1}}u_{i}\overrightarrow{Q_{i}}x_{0}&\textup{(if $x^{\prime}=x_{0}$)}\\[8.53581pt] x\overrightarrow{C_{1}}u_{j}\overleftarrow{Q_{i,j}}u_{i}\overleftarrow{C_{1}}u_{j}^{+}&\textup{(if $x^{\prime}=u_{j}^{+}$)}\end{array}\right..$$

$$x{x}^{\prime }\notin E(G)\text{ and }{d}_G(x;H\cup x\overrightarrow{C_1}{u}_i)+d_G(x^{\prime };H\cup x\overrightarrow{C_1}{u}_i)\le |H\cup x\overrightarrow{C_1}{u}_i|-1.$$

$$d_G(x;H\cup C_1)+d_G(x^{\prime };H\cup C_1)$$

$$=d_G(x;H\cup x\overrightarrow{C_1}{u}_i)+d_G(x^{\prime };H\cup x\overrightarrow{C_1}{u}_i)+d_G(x;u_i^{+}\overrightarrow{C_1}{x}^{-})$$

$$\displaystyle\leq\big{(}|H\cup x\overrightarrow{C_{1}}u_{i}|-1\big{)}+|u_{i}^{+}\overrightarrow{C_{1}}x^{-}|=|H\cup C_{1}|-1.$$

$$X=\{x_0\}\cup \{u_i^{+}:1\le i\le l\}.$$

$$\displaystyle d_{G}(x;C^{*}-C_{1})+d_{G}(x^{\prime};C^{*}-C_{1})\geq n-\big{(}|H\cup C_{1}|-1\big{)}=|C^{*}-C_{1}|+1.$$

$$Y=\{x_0,x_1\}\cup \{u_i^{-}:2\le i\le l\}.$$

$$\displaystyle d_{G}(y;C^{*}-C_{1})+d_{G}(y^{\prime};C^{*}-C_{1})\geq n-\big{(}|H\cup C_{1}|-1\big{)}=|C^{*}-C_{1}|+1.$$

$$d_G(u_1^{+};C^{\ast }-C_1)+d_G(u_i^{-};C^{\ast }-C_1)>|C^{\ast }-C_1|=\sum _{2\le p\le k}|C_p|.$$

$$d_G(u_1^{+};C_2)+d_G(u_i^{-};C_2)\ge |C_2|+1.$$

$$D_1=u_i\overrightarrow{C_1}{u}_1\overrightarrow{Q_{1,i}}{u}_i\text{ and }{D}_2=u_1^{+}\overrightarrow{C_1}{u}_i^{-}{u}^{+}\overrightarrow{C_2}u{u}_1^{+}\text{ (see Figure reffig:insertiblepath)}.$$

Full text

Keywords: Hamilton cycles, 2-factors, Vertex-disjoint cycles, Degree sum conditions

AMS Subject Classification: 05C70, 05C45, 05C38

1 Introduction

In this paper, we consider finite simple graphs, which have neither loops nor multiple edges. For terminology and notation not defined in this paper, we refer the readers to [4]. The independence number and the connectivity of a graph $G$ are denoted by $\alpha(G)$ and $\kappa(G)$, respectively. For a vertex $x$ of a graph $G$, we denote by $d_{G}(x)$ and $N_{G}(x)$ the degree and the neighborhood of $x$ in $G$. Let $\sigma_{m}(G)$ be the minimum degree sum of an independent set of $m$ vertices in a graph $G$, i.e., if $\alpha(G)\geq m$, then

[TABLE]

otherwise, $\sigma_{m}(G)=+\infty$. If the graph $G$ is clear from the context, we often omit the graph parameter $G$ in the graph invariant. In this paper, “disjoint” always means “vertex-disjoint”.

A graph having a hamilton cycle, i.e., a cycle containing all the vertices of the graph, is said to be hamiltonian. The hamiltonian problem has long been fundamental in graph theory. But, it is NP-complete, and so no easily verifiable necessary and sufficient condition seems to exist. Therefore, many researchers have focused on “better” sufficient conditions for graphs to be hamiltonian (see a survey [14]). In particular, the following degree sum condition, due to Ore (1960), is classical and well known.

Theorem A** **(Ore [15])

Let $G$ be a graph of order $n\geq 3$. If $\sigma_{2}\geq n$, then $G$ is hamiltonian.

Chvátal and Erdős (1972) discovered the relationship between the connectivity, the independence number and the hamiltonicity.

Theorem B** **(Chvátal, Erdős [8])

Let $G$ be a graph of order at least $3$. If $\alpha\leq\kappa$, then $G$ is hamiltonian.

Bondy [2] pointed out that the graph satisfying the Ore condition also satisfies the Chvátal-Erdős condition, that is, Theorem B implies Theorem A.

By Theorem B, we should consider the degree condition for the existence of a hamilton cycle in graphs $G$ with $\alpha(G)\geq\kappa(G)+1$. In fact, Bondy (1980) gave the following degree sum condition by extending Theorem B.

Theorem C** **(Bondy [3])

Let $G$ be an $m$-connected graph of order $n\geq 3$. If $\sigma_{m+1}>\frac{1}{2}(m+1)(n-1)$, then $G$ is hamiltonian.

In 2008, Yamashita [17] introduced a new graph invariant and further generalized Theorem C as follows. For a vertex subset $X$ of a graph $G$ with $|X|\geq t$, we define

[TABLE]

Let $m\geq t$, and if $\alpha(G)\geq m$, then let

[TABLE]

otherwise, $\sigma_{t}^{m}(G)=+\infty$. Note that $\sigma_{t}^{m}(G)\geq\frac{t}{m}\cdot\sigma_{m}(G)$.

Theorem D** **(Yamashita [17])

Let $G$ be an $m$-connected graph of order $n\geq 3$. If $\sigma_{2}^{m+1}\geq n$, then $G$ is hamiltonian.

This result suggests that the degree sum of non-adjacent “two” vertices is important for hamilton cycles.

On the other hand, it is known that a 2-factor is one of the important generalizations of a hamilton cycle. A 2-factor of a graph is a spanning subgraph in which every component is a cycle, and thus a hamilton cycle is a 2-factor with “exactly 1 cycle”. As one of the studies concerning the difference between hamilton cycles and 2-factors, in this paper, we focus on 2-factors with “exactly $k$ cycles”. Similar to the situation for hamilton cycles, deciding whether a graph has a 2-factor with $k\ (\geq 2)$ cycles is also NP-complete. Therefore, the sufficient conditions for the existence of such a 2-factor also have been extensively studied in graph theory (see a survey [11]). In particular, the following theorem, due to Brandt, Chen, Faudree, Gould and Lesniak (1997), is interesting. (In the paper [5], the order condition is not “$n\geq 4k-1$” but “$n\geq 4k$”. However, by using a theorem of Enomoto [9] and Wang [16] (“every graph $G$ of order at least $3k$ with $\sigma_{2}(G)\geq 4k-1$ contains $k$ disjoint cycles”) for the cycles packing problem, we can obtain the following. See the proof in [5, Lemma 1].)

Theorem E** **(Brandt et al. [5])

Let $k$ be a positive integer, and let $G$ be a graph of order $n\geq 4k-1$. If $\sigma_{2}\geq n$, then $G$ has a 2-factor with exactly $k$ cycles.

This theorem shows that the Ore condition guarantees the existence of a hamilton cycle but also the existence of a 2-factor with a prescribed number of cycles.

By considering the relation between Theorem A and Theorem E, Chen, Gould, Kawarabayashi, Ota, Saito and Schiermeyer [6] conjectured that the Chvátal-Erdős condition in Theorem B also guarantees the existence of a 2-factor with exactly $k$ cycles (see [6, Conjecture 1]). Chen et al. also proved that if the order of a 2-connected graph $G$ with $\alpha(G)=\alpha\leq\kappa(G)$ is sufficiently large compared with $k$ and with the Ramsey number $r(\alpha+4,\alpha+1)$, then the graph $G$ has a 2-factor with $k$ cycles. In [12], Kaneko and Yoshimoto “almost” solved the above conjecture for $k=2$ (see the comment after Theorem E in Chen et al. [6] for more details). Another related result can be found in [7]. But, the above conjecture is still open in general. In this sense, there is a big gap between hamilton cycles and 2-factors with exactly $k\ (\geq 2)$ cycles.

In this paper, by combining the techniques of the proof for hamiltonicity and the proof for 2-factors with a prescribed number of cycles, we give the following Yamashita-type condition for 2-factors with $k$ cycles.

Theorem 1

Let $k$ be a positive integer, and let $G$ be an $m$-connected graph of order $n\geq 5k-2$. If $\sigma_{2}^{\lceil m/k\rceil+1}\geq n$, then $G$ has a 2-factor with exactly $k$ cycles.

This theorem implies the following:

Remark 2

• •

Theorem 1 is a generalization of Theorem D.

• •

Theorem 1 leads to the Bondy-type condition: If $G$ is an $m$-connected graph of order $n\geq 5k-2$ with $\sigma_{\lceil m/k\rceil+1}(G)>\frac{1}{2}(\lceil m/k\rceil+1)(n-1)$, then $G$ has a 2-factor with exactly $k$ cycles. Therefore, Theorem 1 is also a generalization of Theorem E for sufficiently large graphs. (Recall that $\sigma_{t}^{m}(G)\geq\frac{t}{m}\cdot\sigma_{m}(G)$ and $\sigma_{m}(G)\geq\frac{m}{2}\cdot\sigma_{2}(G)$ for $m\geq t\geq 2$.)

• •

Theorem 1 leads to the Chvátal-Erdős-type condition: If $G$ is a graph of order at least $5k-2$ with $\alpha(G)\leq\lceil\kappa(G)/k\rceil$, then $G$ has a 2-factor with exactly $k$ cycles.

The complete bipartite graph $K_{(n-1)/2,(n+1)/2}$ ($n$ is odd) does not contain a 2-factor, and hence the degree condition in Theorem 1 is best possible in this sense. The order condition in Theorem 1 comes from our proof techniques. Similar to the situation for the proof of Theorem E, we will use the order condition only for the cycles packing problem (see Lemma 5 and the proof of Theorem 1 in Section 3). The complete bipartite graph $K_{2k-1,2k-1}$ shows that $n\geq 4k-1$ is necessary. In the last section (Section 4), we note that “$n\geq 5k-2$” can be replaced with “$n\geq 4k-1$” for the Bondy-type condition (and the Chvátal-Erdős-type condition) in Remark 2.

Table 1 summarizes the conditions mentioned in the above.

To prove Theorem 1, in the next section, we extend the concept of insertible vertices which was introduced by Ainouche [1], and we prove Theorem 1 in Section 3 by using it.

2 The concept of insertible vertices

In this section, we prepare terminology and notations and give some lemmas.

Let $G$ be a graph. For $v\in V(G)$ and $X\subseteq V(G)$, we let $N_{G}(v;X)=N_{G}(v)\cap X$ and $d_{G}(v;X)=|N_{G}(v;X)|$. For $V,X\subseteq V(G)$, let $N_{G}(V;X)=\bigcup_{v\in V}N_{G}(v;X)$. For $X\subseteq V(G)$, we denote by $G[X]$ the subgraph of $G$ induced by $X$. An $(x,y)$-path in $G$ is a path from a vertex $x$ to a vertex $y$ in $G$ . We write a cycle (or a path) $C$ with a given orientation by $\overrightarrow{C}$. If there exists no fear of confusion, we abbreviate $\overrightarrow{C}$ by $C$. Let $C$ be an oriented cycle (or path). We denote by $\overleftarrow{C}$ the cycle $C$ with a reverse orientation. For $x\in V(C)$, we denote the successor and the predecessor of $x$ on $\overrightarrow{C}$ by $x^{+}$ and $x^{-}$. For $x,y\in V(C)$, we denote by $x\overrightarrow{C}y$ the $(x,y)$-path on $\overrightarrow{C}$. The reverse sequence of $x\overrightarrow{C}y$ is denoted by $y\overleftarrow{C}x$. In the rest of this paper, we consider that every cycle (path) has a fixed orientation, unless stated otherwise, and we often identify a subgraph $H$ of $G$ with its vertex set $V(H)$.

The following lemma is obtained by using the standard crossing argument, and so we omit the proof.

Lemma 1

Let $G$ be a graph of order $n$, and let $P$ be an $(x,y)$-path of order at least $3$ in $G$. If $d_{G}(x)+d_{G}(y)\geq n$, then $G$ contains a cycle of order at least $|P|$.

In [1], Ainouche introduced the concept of insertible vertices, which has been used for the proofs of the results on hamilton cycles. In this paper, we modify it for 2-factors with $k$ cycles, and it also plays a crucial role in our proof. Let $G$ be a graph, and let $\mathcal{D}=\{D_{1},\dots,D_{r+s}\}$ $(r+s\geq 1)$ be the set of $r$ cycles and $s$ paths in $G$ which are pairwise disjoint. For a vertex $x$ in $G-\bigcup_{1\leq p\leq r+s}D_{p}$, the vertex $x$ is insertible for $\mathcal{D}$ if there is an edge $uv$ in $E(D_{p})$ such that $xu,xv\in E(G)$ for some $p$ with $1\leq p\leq r+s$. In the following lemma, “partition” of a graph means a partition of the vertex set.

Lemma 2

Let $G$ be a graph, and let $\mathcal{D}=\{D_{1},\dots,D_{r+s}\}$ $(r+s\geq 1)$ be the set of $r$ cycles and $s$ paths in $G$ which are pairwise disjoint, and $P$ be a path in $G-\bigcup_{1\leq p\leq r+s}D_{p}$. If every vertex of $P$ is insertible for $\mathcal{D}$, then G\big{[}\bigcup_{1\leq p\leq r+s}V(D_{p})\cup V(P)\big{]} can be partitioned into $r$ cycles and $s$ paths.

Proof of Lemma 2. By choosing the following two vertices $u,v\in V(P)$ and the edge $ww^{+}\in\bigcup_{1\leq p\leq r+s}E(D_{p})$ inductively, we can get the desired partition of G\big{[}\bigcup_{1\leq p\leq r+s}V(D_{p})\cup V(P)\big{]}. Let $u$ be the first vertex along $\overrightarrow{P}$, and take an edge $ww^{+}$ in E(D_{i})\ \big{(}\subseteq\bigcup_{1\leq p\leq r+s}E(D_{p})\big{)} such that $uw,uw^{+}\in E(G)$ for some $i$ with $1\leq i\leq r+s$ (since $u$ is insertible for $\mathcal{D}$, we can take such an edge). We let $v$ be the last vertex along $\overrightarrow{P}$ such that $vw,vw^{+}\in E(G)$ (may be $u=v$). Then, we can insert all vertices of $u\overrightarrow{P}v$ into $D_{i}$. In fact, by replacing the edge $ww^{+}$ by the path $wu\overrightarrow{P}vw^{+}$, we can obtain a spanning subgraph $D_{i}^{\prime}$ of $G[V(D_{i}\cup u\overrightarrow{P}v)]$ such that $D_{i}^{\prime}$ is a cycle if $D_{i}$ is a cycle; otherwise, $D_{i}^{\prime}$ is a path. By the choice of $u$ and $v$, we have $zw\notin E(G)$ or $zw^{+}\notin E(G)$ for each vertex $z$ of $P^{\prime}:=P-u\overrightarrow{P}v$, and hence every vertex of $P^{\prime}$ is insertible for $\mathcal{D^{\prime}}=\{D_{1},\dots,D_{i-1},D_{i}^{\prime},D_{i+1},\dots,D_{r+s}\}$. Thus, we can repeat this argument for the path $P^{\prime}$ and the set $\mathcal{D}^{\prime}$, and we get then the desired partition. ∎

In the rest of this section, we fix the following. Let $C_{1},\dots,C_{k}$ be $k$ disjoint cycles in a graph $G$, and let $C^{*}=\bigcup_{1\leq p\leq k}C_{p}$. Choose $C_{1},\dots,C_{k}$ so that

[TABLE]

Suppose that $C^{*}$ does not form a 2-factor of $G$. Let $H=G-C^{*}$, and let $H_{0}$ be a component of $H$ and $x_{0}\in V(H_{0})$. Let

[TABLE]

We assume that $u_{1},u_{2},\dots,u_{l}$ appear in this order on $\overrightarrow{C_{1}}$, and let $u_{l+1}=u_{1}$. Note that by the maximality of $|C^{*}|$, $u_{i}^{+}\neq u_{i+1}$ for $1\leq i\leq l$. We denote by $\overrightarrow{Q_{i}}$ and $\overrightarrow{Q_{i,j}}$ a $(u_{i},x_{0})$-path in $G[V(H_{0})\cup\{u_{i}\}]$ and a $(u_{i},u_{j})$-path passing through a vertex of $H_{0}$ in $G[V(H_{0})\cup\{u_{i},u_{j}\}]$, respectively.

Lemma 3

For $1\leq i\leq l$, $u_{i}^{+}\overrightarrow{C_{1}}u_{i+1}^{-}$ contains a non-insertible vertex for $\{C_{2},\dots,C_{k}\}$.

Proof of Lemma 3. Suppose that every vertex of $u_{i}^{+}\overrightarrow{C_{1}}u_{i+1}^{-}$ is insertible for $\{C_{2},\dots,$ $C_{k}\}$. Then, by Lemma 2, G\big{[}\bigcup_{2\leq p\leq k}V(C_{p})\cup V(u_{i}^{+}\overrightarrow{C_{1}}u_{i+1}^{-})\big{]} has a 2-factor with exactly $k-1$ cycles. With the cycle $u_{i+1}\overrightarrow{C_{1}}u_{i}\overrightarrow{Q_{i,i+1}}u_{i+1}$, we can get $k$ disjoint cycles in $G$ such that the sum of the orders is larger than $|C^{*}|$, a contradiction. ∎

For $1\leq i\leq l$, let $x_{i}$ be the first non-insertible vertex for $\{C_{2},\dots,C_{k}\}$ in $V(u_{i}^{+}\overrightarrow{C_{1}}u_{i+1}^{-})$ on $\overrightarrow{C_{1}}$, i.e., every vertex of $u_{i}^{+}\overrightarrow{C_{1}}x_{i}^{-}$ is insertible for $\{C_{2},\dots,C_{k}\}$, but $x_{i}$ is not insertible (Lemma 3 guarantees the existence of such a vertex $x_{i}$).

Lemma 4

Let $i,j$ be integers with $1\leq i,j\leq l$ and $i\neq j$. If $x\in V(u_{i}^{+}\overrightarrow{C_{1}}x_{i})$ and $x^{\prime}\in\{x_{0},u_{j}^{+}\}$, then (i) $xx^{\prime}\notin E(G)$, and (ii) $d_{G}(x;H\cup C_{1})+d_{G}(x^{\prime};H\cup C_{1})\leq|H\cup C_{1}|-1$.

Proof of Lemma 4. Consider the path

[TABLE]

See Figure 1. Then, $P$ is a path in $G[V(H\cup x\overrightarrow{C_{1}}u_{i})]$ passing through all vertices of $x\overrightarrow{C_{1}}u_{i}$ and a vertex of $H_{0}$. Recall that every vertex of $u_{i}^{+}\overrightarrow{C_{1}}x^{-}$ is insertible for $\{C_{2},\dots,C_{k}\}$, and hence G\big{[}\bigcup_{2\leq p\leq k}V(C_{p})\cup V(u_{i}^{+}\overrightarrow{C_{1}}x^{-})\big{]} has a 2-factor with exactly $k-1$ cycles (by Lemma 2). Hence, the maximality of $|C^{*}|$ and Lemma 1 yield that

[TABLE]

In particular, (i) holds. Then, by applying (i) for each vertex in $u_{i}^{+}\overrightarrow{C_{1}}x^{-}$ and the vertex $x^{\prime}$, we have $N_{G}(x^{\prime};u_{i}^{+}\overrightarrow{C_{1}}x^{-})=\emptyset$. Combining this with the above inequality, we get,

[TABLE]

Thus (ii) also holds. ∎

3 Proof of Theorem 1

Before proving Theorem 1, we will give the following lemma for the cycles packing problem.

Lemma 5

Let $k,m,n$ and $G$ be the same ones as in Theorem 1. Under the same degree sum condition as Theorem 1, $G$ contains $k$ disjoint cycles.

Proof of Lemma 5. If $k=1$, then it is easy to check that $G$ contains a cycle. If $\lceil m/k\rceil=1$ or $\lceil m/k\rceil\geq 3$, then by a theorem of Enomoto [9], $G$ contains $k$ disjoint cycles (note that if $\lceil m/k\rceil\geq 3$, then $G$ is $(2k+1)$-connected, that is, the minimum degree $\delta(G)$ is at least $2k+1$). Thus, we may assume that $k\geq 2$ and $\lceil m/k\rceil=2$. Then, we have $\delta(G)\geq m\geq k+1$ and $\sigma_{2}^{3}(G)=\sigma_{2}^{\lceil m/k\rceil+1}(G)\geq n\geq 5k-2$. Note that, by the definition of $\sigma_{2}^{3}(G)$ and $\sigma_{3}(G)$, $\sigma_{3}(G)\geq\sigma_{2}^{3}(G)+\delta(G)$. Note also that $n\geq 5k-2\geq 3k+2\geq 8$ because $k\geq 2$. Hence, by a theorem of Fujita et al. [10] (“every graph $G$ of order at least $3k+2\geq 8$ with $\sigma_{3}(G)\geq 6k-2$ contains $k$ disjoint cycles”), we can get the desired conclusion. ∎

Now we are ready to prove Theorem 1.

Proof of Theorem 1. Let $G$ be an $m$-connected graph of order $n\geq 5k-2$ such that $\sigma_{2}^{\lceil m/k\rceil+1}(G)\geq n$. We show that $G$ has a 2-factor with exactly $k$ cycles. By Theorem E, we may assume that $\lceil m/k\rceil\geq 2$. By Lemma 5, $G$ contains $k$ disjoint cycles. Let $C_{i}$ ($1\leq i\leq k$), $C^{*}$, $H$, $H_{0}$, $x_{0}$ and $u_{i}$ ($1\leq i\leq l$) be the same graphs and vertices as the ones described in the paragraph preceding Lemma 3 in Section 2. In particular, we may assume that $l=\lceil m/k\rceil$. Because, since $G$ is $m$-connected, it follows that $|N_{G}(H_{0};C^{*})|\geq m$ (note that by the maximality of $|C^{*}|$, $|C^{*}|>m$), and hence, without loss of generality, we may assume that $|N_{G}(H_{0};C_{1})|\geq\lceil m/k\rceil\ (\geq 2)$.

We first consider the set

[TABLE]

Then, Lemma 4 implies the following:

1. (1)

$X$ is an independent set of size $l+1$. 2. (2)

$d_{G}(x;H\cup C_{1})+d_{G}(x^{\prime};H\cup C_{1})\leq|H\cup C_{1}|-1$ for $x,x^{\prime}\in X$ ($x\neq x^{\prime}$).

On the other hand, by the maximality of $|C^{*}|$ and Lemma 2, $x_{0}$ is non-insertible for $\{C_{2},\dots,C_{k}\}$. This implies the following:

1. (3)

$d_{G}(x_{0};C_{p})\leq|C_{p}|/2$ for $2\leq p\leq k$, and hence $d_{G}(x_{0};C^{*}-C_{1})\leq|C^{*}-C_{1}|/2$.

Since $\sigma_{2}^{l+1}(G)\geq n$, it follows from (1) that there exist two distinct vertices $x$ and $x^{\prime}$ in $X$ such that $d_{G}(x)+d_{G}(x^{\prime})\geq n$. Then, by (2), we get

[TABLE]

Combining this with (3) and the definition of $X$, we may assume that

1. (4)

$d_{G}(u_{1}^{+};C^{*}-C_{1})>|C^{*}-C_{1}|/2$.

Next, let $x_{1}$ be the first non-insertible vertex for $\{C_{2},\dots,C_{k}\}$ in the path $u_{1}^{-}\overleftarrow{C_{1}}u_{l}^{+}$ on $\overleftarrow{C_{1}}$ (we can take such a vertex by Lemma 3 and the symmetry of $\overrightarrow{C_{1}}$ and $\overleftarrow{C_{1}}$), and we consider the set

[TABLE]

Then, by the symmetry of $\overrightarrow{C_{1}}$ and $\overleftarrow{C_{1}}$, Lemma 4, and since $x_{1}$ is non-insertible for $\{C_{2},\dots,C_{k}\}$, we have the following:

1. (5)

$Y$ is an independent set of size $l+1$. 2. (6)

$d_{G}(y;H\cup C_{1})+d_{G}(y^{\prime};H\cup C_{1})\leq|H\cup C_{1}|-1$ for $y,y^{\prime}\in Y$ ($y\neq y^{\prime}$). 3. (7)

$d_{G}(x_{1};C_{p})\leq|C_{p}|/2$ for $2\leq p\leq k$, and hence $d_{G}(x_{1};C^{*}-C_{1})\leq|C^{*}-C_{1}|/2$.

Since $\sigma_{2}^{l+1}(G)\geq n$, it follows from (5) that there exist two distinct vertices $y$ and $y^{\prime}$ in $Y$ such that $d_{G}(y)+d_{G}(y^{\prime})\geq n$. Then, by (6), we get

[TABLE]

Combining this with (3), (7) and the definition of $Y$, we have the following:

1. (8)

$d_{G}(u_{i}^{-};C^{*}-C_{1})>|C^{*}-C_{1}|/2$ for some $i$ with $2\leq i\leq l$.

By (4) and (8), we have

[TABLE]

Hence, there exists a cycle $C_{p}$ ($2\leq p\leq k$), say $p=2$, such that

[TABLE]

This implies that there exists an edge $uv$ in $E(C_{2})$ such that $u_{1}^{+}u,u_{i}^{-}v\in E(G)$. By changing the orientation of $C_{2}$ if necessary, we may assume that $u^{+}=v$. Note that $i\geq 2$, and consider two cycles

[TABLE]

Then, $D_{1},D_{2},C_{3},\dots,C_{k}$ are $k$ disjoint cycles such that the sum of the orders is larger than $|C^{*}|$, a contradiction.

This completes the proof of Theorem 1. ∎

4 Notes on the order condition

As shown in the argument of the previous section, in the proof of Theorem 1, the order condition “$n\geq 5k-2$” is required only to show the existence of $k$ disjoint cycles in a graph $G$ (recall that the order condition in Theorem E is also). Therefore, the proof of Theorem 1 actually implies the following.

Theorem 3

Let $k$ be a positive integer, and let $G$ be an $m$-connected graph of order $n$. Suppose that $G$ contains $k$ disjoint cycles. If $\sigma_{2}^{\lceil m/k\rceil+1}\geq n$, then $G$ has a 2-factor with exactly $k$ cycles.

From this theorem, if we can obtain better results on the cycles packing problem, then the order conditions in Theorem 1 and Remark 2 can be improved. In fact, by using the result of Kierstead, Kostochka and Yeager (2017) and modifying the proof of Lemma 5, we can obtain a sharp order condition for the result in Remark 2 (see Corollary 4).

Theorem F** **(Kierstead et al. [13])

Let $k$ be an integer with $k\geq 2$, and let $G$ be a graph of order $n\geq 3k$ with $\delta(G)\geq 2k-1$. Then $G$ contains $k$ disjoint cycles if and only if (i) $\alpha(G)\leq n-2k$, and (ii) if $k$ is odd and $n=3k$, then $G\not\cong 2K_{k}\vee\overline{K_{k}}$ and if $k=2$, then $G$ is not a wheel.

Lemma 6

Let $k$ be a positive integer, and let $G$ be an $m$-connected graph of order $n\geq 4k-1$. If $\sigma_{\lceil m/k\rceil+1}(G)>\frac{1}{2}(\lceil m/k\rceil+1)(n-1)$, then $G$ contains $k$ disjoint cycles.

Proof of Lemma 6. By a similar argument as in the proof of Lemma 5, we have the following: If $k=1$, then we can easily find a cycle; If $\lceil m/k\rceil=1$ or $\lceil m/k\rceil\geq 3$, then by a theorem of Enomoto [9], $G$ contains $k$ disjoint cycles; If $\lceil m/k\rceil=2$, and $k\geq 3$ or $n\geq 4k$, then by a theorem of Fujita et al. [10], $G$ contains $k$ disjoint cycles. Thus, we may assume that $k=2$, $\lceil m/k\rceil=2$ and $n=4k-1=7$. Then, $\delta(G)\geq m\geq k+1=3=2k-1$ and $\sigma_{3}(G)>\frac{3}{2}(n-1)=6k-3=9$. Since $n=7$ and $\sigma_{3}(G)>9$, it follows that $\alpha(G)\leq 3=n-2k$ and $G$ is not a wheel. Hence, by Theorem F, $G$ contains two disjoint cycles. Thus, the lemma follows. ∎

Recall that $\sigma_{t}^{m}(G)\geq\frac{t}{m}\cdot\sigma_{m}(G)$ for $m\geq t\geq 2$, and hence Theorem 3 and Lemma 6 lead to the following.

Corollary 4

Let $k$ be a positive integer, and let $G$ be an $m$-connected graph of order $n\geq 4k-1$. If $\sigma_{\lceil m/k\rceil+1}(G)>\frac{1}{2}(\lceil m/k\rceil+1)(n-1)$, then $G$ has a 2-factor with exactly $k$ cycles.