Equitable factorizations of edge-connected graphs

TL;DR

This paper establishes conditions under which highly edge-connected graphs can be decomposed into factors with degrees close to a uniform distribution, with applications to parity and degree constraints.

Contribution

It introduces new edge-decomposition theorems for highly edge-connected graphs with degree and parity constraints, extending previous results in graph factorization.

Findings

Edge-connected graphs can be decomposed into factors with degrees close to uniform.

Specific conditions allow for parity-preserving decompositions with degree bounds.

A sufficient condition for parity factors with prescribed degree deviations is provided.

Abstract

In this paper, we show that every $(3k-3)$-edge-connected graph $G$, under a certain condition on whose degrees, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$, $|d_{G_i}(v)-d_G(v)/k|< 1$, where $1\le i\le k$. As application, we deduce that every $6$-edge-connected graph $G$ can be edge-decomposed into three factors $G_1$, $G_2$, and $G_3$ such that for each vertex $v\in V(G_i)$, $|d_{G_i}(v)-d_{G}(v)/3|< 1$, unless $G$ has exactly one vertex $z$ with $d_G(z) \stackrel{3}{\not\equiv}0$. Next, we show that every odd-$(3k-2)$-edge-connected graph $G$ can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$, $d_{G_i}(v)$ and $d_G(v)$ have the same parity and $|d_{G_i}(v)-d_G(v)/k|< 2$, where $k$ is an odd positive integer and $1\le i\le k$. Finally, we give a sufficient edge-connectivity condition for a graph $G$ to have a parity factor $F$ with specified odd-degree vertices such that for each vertex $v$, $| d_{F}(v)-\varepsilon d_G(v)|< 2$, where $\varepsilon $ is a real number with $0< \varepsilon < 1$.

Equations

$$|d_{G_i}(v)-d_G(v)/k|<1.$$

$$|d_{G_i}(v)-d_G(v)/k|\le 1.$$

$$|d_{G_i}(v)-d_G(v)/k|<1.$$

$$|d_{G_i}(v)-d_G(v)/k|<2.$$

$$|d_F(v)-\epsilon d_G(v)|<2.$$

$$|d_{G_i}^{+}(v)-d_G^{+}(v)/k|<1\text{ and }|d_{G_i}^{-}(v)-d_G^{-}(v)/k|<1.$$

$$\lfloor \frac{d_G(v)}{2k}\rfloor +\lfloor \frac{d_G(v)+1}{2k}\rfloor \le d_{G_i}(v)\le \lceil \frac{d_G(v)}{2k}\rceil +\lceil \frac{d_G(v)-1}{2k}\rceil .$$

$$|d_{G_i}(v)-d_G(v)/k|<\{\begin{array}{ll}2& \text{when v=z};\\ 1,& \text{otherwise}.\end{array}$$

$$|d_{G_i}(v)-d_G(v)/k|<1.$$

$$|d_{G_i}(v)-d_G(v)/3|<1.$$

$$|d_{G_i}(v)-d_G(v)/k|<2.$$

$$d_G(X)\ge 3k-2,$$

$$d_G(X)\ge 3k-3,$$

$$|d_{G_i}(v)-{\epsilon }_i{d}_G(v)|<3.$$

$${\omega }_f(G,A,B)<2+\sum _{v\in A}f(v)+\sum _{v\in B}(d_G(v)-g(v))-d_G(A,B),$$

$$d_G(X)\ge \{\begin{array}{ll}1/\epsilon ,& \text{when ∑v∈Xf(v) is odd};\\ 1/(1-\epsilon ),& \text{when ∑v∈X(dG(v)−f(v)) is odd},\end{array}$$

$$\lfloor \epsilon d_G(v)\rfloor -1\le d_F(v)\le \lceil \epsilon d_G(v)\rceil +1.$$

$$\sum _{v\in A}\epsilon d_G(v)+\sum _{v\in B}(1-\epsilon )d_G(v)<2+\sum _{v\in A}{f}_0(v)+\sum _{v\in B}(d_G(v)-g_0(v)),$$

$$\sum _{X\in P}(\epsilon d_G(X,A)+(1-\epsilon )d_G(X,B))\le \sum _{v\in A}\epsilon d_G(v)+\sum _{v\in B}(1-\epsilon )d_G(v)-d_G(A,B).$$

$$|P_c|=\sum _{X\in P_c}(\epsilon +(1-\epsilon ))\le \sum _{X\in P_c}(\epsilon d_G(X,A)+(1-\epsilon )d_G(X,B)).$$

$$\sum _{v\in X}f(v)\stackrel{2}{\not\equiv }{d}_G(X,B)=d_G(X)\stackrel{2}{\equiv }\sum _{v\in X}{d}_G(v),$$

$$|P_a|\le \sum _{X\in P_a}(1-\epsilon )d_G(X)=\sum _{X\in P_a}(\epsilon d_G(X,A)+(1-\epsilon )d_G(X,B)),$$

$$|P_b|\le \sum _{X\in P_b}\epsilon d_G(X)=\sum _{X\in P_b}(\epsilon d_G(X,A)+(1-\epsilon )d_G(X,B)).$$

$$|P|=|P_a|+|P_b|+|P_c|\le \sum _{X\in P}(\epsilon d_G(X,A)+(1-\epsilon )d_G(X,B)).$$

$${\omega }_f(G,A,B)=|P|<2+\sum _{v\in A}{f}_0(v)+\sum _{v\in B}(d_G(v)-g_0(v))-d_G(A,B).$$

$$\lfloor \frac{d_G(v)}{2}\rfloor -1\le d_F(v)\le \lceil \frac{d_G(v)}{2}\rceil +1.$$

Full text

*Keywords: Factorization; equitable edge-coloring; odd-edge-connectivity; modulo orientation; regular graph. *

1 Introduction

In this article, graphs may have loops and multiple edges. Let $G$ be a graph. The vertex set, the edge set, and the maximum degree of $G$ are denoted by $V(G)$, $E(G)$, and, $\Delta(G)$, respectively. We denote by $d_{G}(v)$ the degree of a vertex $v$ in the graph $G$, whether $G$ is directed or not. Also the out-degree and in-degree of $v$ in a directed graph $G$ are denoted by $d_{G}^{+}(v)$ and $d_{G}^{-}(v)$. An orientation of $G$ is said to be $p$-orientation, if for each vertex $v$, $d_{G}^{+}(v)\stackrel{{\scriptstyle k}}{{\equiv}}p(v)$, where $p:V(G)\rightarrow Z_{k}$ is a mapping and $Z_{k}$ is the cyclic group of order $k$. For any two disjoint vertex sets $A$ and $B$, we denote by $d_{G}(A,B)$ the number of edges with one end in $A$ and the other one in $B$. Also, we denote by $d_{G}(A)$ the number of edges of $G$ with exactly one end in $A$. Note that $d_{G}(\{v\})$ denotes the number of non-loop edges incident with $v$, while $d_{G}(v)$ denotes the degree of $v$. We denote by $G[A]$ the induced subgraph of $G$ with the vertex set $A$ containing precisely those edges of $G$ whose ends lie in $A$. Let $g$ and $f$ be two integer-valued functions on $V(G)$. A parity $(g,f)$-factor of $G$ refers to a spanning subgraph $F$ such that for each vertex $v$, $g(v)\leq d_{F}(v)\leq f(v)$, and also $d_{F}(v)$, $g(v)$, and $f(v)$ have the same parity. An $f$-parity factor refers to a spanning subgraph $F$ such that for each vertex $v$, $d_{F}(v)$ and $f(v)$ have the same parity. A graph $G$ is called $m$-tree-connected, if it contains $m$ edge-disjoint spanning trees. A graph $G$ is termed essentially $\lambda$-edge-connected, if all edges of any edge cut of size strictly less than $\lambda$ are incident to a vertex. A graph $G$ is called odd-$\lambda$-edge-connected, if $d_{G}(A)\geq\lambda$, for every vertex $A$ with $d_{G}(A)$ odd. Note that $2\lambda$-edge-connected graphs are odd-$(2\lambda+1)$-edge-connected. A factorization $G_{1},\ldots,G_{k}$ of $G$ is called equitable factorization, if for each vertex $v$, $|d_{G_{i}}(v)-d_{G}(v)/k|<1$, where $1\leq i\leq k$. Throughout this article, all variables $k$ are positive and integer.

In 1971 de Werra observed that bipartite graphs admit equitable factorizations.

Theorem 1.1

.([27])* Every bipartite graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,*

[TABLE]

In 1994 Hilton and de Werra developed Theorem 1.1 to simple graphs by replacing the following weaker version. In 2008 Hilton [13] conjectured that for those simple graphs for which whose vertices with degree divisible by $k$ form an induced forest, the upper bound can be replaced by that of Theorem 1.1. Later, this conjecture was confirmed by Zhang and Liu (2011) [28].

Theorem 1.2

.([15])* Every simple graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,*

[TABLE]

Recently, Thomassen (2019) made another attempt for developing Theorem 1.1 to highly edge-connected regular graphs and concluded the following result. An intresting consequence of it says that every $6$-edge-connected $3r$-regular graph can be edge-decomposed into three $r$-regular factors.

Theorem 1.3

.([26])* Let $k$ and $r$ be two odd positive integers. If $G$ is an odd-$(3k-2)$-edge-connected $kr$-regular graph, then it can be edge-decomposed into $r$-regular factors.*

In this paper, we generalize Thomassen’s result in several ways together with developing Theorem 1.1 to highly edge-connected graphs as the next theorem. In addition, we provide a common version for Theorem 1.3 and whose even-regular version by replacing another concept of edge-connectivity.

Theorem 1.4

.* Let $G$ be a $(3k-3)$-edge-connected graph, where $k$ is a positive integer. If there is a vertex set $Z$ with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$, then $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,*

[TABLE]

In 1982 Hilton constructed the following parity version of Theorem 1.1 on the existence of even factorizations with bounded degrees of even graphs.

Theorem 1.5

.([14])* Every graph $G$ with even degrees can be edge-decomposed into $k$ even factors $G_{1},\ldots,G_{k}$ such that for each vertex $v$, $|d_{G_{i}}(v)-d_{G}(v)/k|<2.$*

In this paper, we provide two extensions of Hilton’s result in highly odd-edge-connected graphs and highly edge-connected even graphs by proving the following results. An interesting consequence of them says that every $(3k-3)$-edge-connected graph of even order whose degrees lie in the set $\{3k,3k+2,...,5k\}$ can be edge-decomposed into $k$ factors whose degrees lie in the set $\{3,5\}$. Note that the next theorem can also be considered as an extension of Theorem 1.3.

Theorem 1.6

.* Let $G$ be a graph and let $k$ be an odd positive integer. If $G$ is odd-$(3k-2)$-edge-connected, then it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}d_{G}(v)$ and*

[TABLE]

Theorem 1.7

.* Let $G$ be a graph with even degrees, let $f:V(G)\rightarrow Z_{2}$ be a mapping with $\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$, and let $k$ be an even positive integer. If for every vertex set $X$ with $\sum_{v\in X}f(v)$ odd, $d_{G}(X)\geq 3k-2$, then $G$ can be edge-decomposed into $f$-parity factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,*

[TABLE]

In the remainder of this paper, we turn our attention to the existence of a parity factor whose degrees are close to $\varepsilon$ of the corresponding degrees in the main graph, and we come up with the following results on even graphs.

Theorem 1.8

.* Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If $G$ is a graph with even degrees, then it admits an even factor $F$ such that for each vertex $v$,*

[TABLE]

Theorem 1.9

.* Let $\varepsilon_{1},\ldots,\varepsilon_{k}$ be $k$ positive real numbers with $\varepsilon_{1}+\cdots+\varepsilon_{k}=1$. If $G$ is a graph with even degrees, then it can be edge-decomposed into $k$ even factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $|d_{G_{i}}(v)-\varepsilon_{i}d_{G}(v)|<6.$*

2 Preliminary results on directed graphs

The following theorem provides a relationship between orientations and equitable factorizations of graphs which is motivated by Theorem 2 in [26]. This result plays an important role in the next section.

Theorem 2.1

.* Every directed graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$ that $1\leq i\leq k$,*

[TABLE]

In particular, $|d_{G_{i}}(v)-d_{G}(v)/k|<1$, when $d^{+}_{G}(v)$ or $d^{-}_{G}(v)$ is divisible by $k$. Furthermore, one can impose at least one of the following conditions arbitrary:

1. $(i)$

$|\,|E(G_{i})|-|E(G)|/k\,|<1$, where $1\leq i\leq k$. 2. $(ii)$

If $u\in V(G)$ and $d_{G}^{+}(u)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(u)$, then $d_{G_{i}}(u)\stackrel{{\scriptstyle 2}}{{\equiv}}(d_{G}^{+}(u)-d_{G}^{-}(u))/k$, where $1\leq i\leq k$.

**Proof. **

The proof presented here is inspired by the proof of Theorem 2 in [26]. First split every vertex $v$ into two vertices $v^{+}$ and $v^{-}$ such that $v^{+}$ whose incident edges were directed away from $v$ in $G$ and $v^{-}$ whose incident edges were directed toward $v$ in $G$. In this construction, every loop in $G$ incident with $v$ is transformed into an edge between $v^{+}$ and $v^{-}$. Now, split every vertex in the new graph into vertices with degrees divisible by $k$, except possibly one vertex with degree less than $k$. Call the resulting (loopless) bipartite graph $G_{0}$. Since $G_{0}$ has maximum degree at most $k$, it admits a proper edge-coloring with at most $k$ colors $c_{1}\ldots,c_{k}$ (to prove this, it is enough to consider a $k$-regular supergraph of it and apply König Theorem [19]). Let $G_{i}$ be the factor of $G$ consisting of the edges of $G$ corresponding to the edges with color $c_{i}$ in $G_{0}$. Note that for every vertex $v$, at least $\lfloor d^{+}_{G}(v)/k\rfloor$ (resp. $\lfloor d^{-}_{G}(v)/k\rfloor$) edges with color $c_{i}$ are incident with the vertex $v^{+}$ (resp. $v^{-}$) in $G_{0}$. Moreover, at most $\lceil d^{+}_{G}(v)/k\rceil$ (resp. $\lceil d^{-}_{G}(v)/k\rceil$) edges with color $c_{i}$ are incident with the vertex $v^{+}$ (resp. $v^{-}$) in $G_{0}$. This completes the proof of the first part.

First we are going to prove item (i). Let us consider such a proper edge-coloring with the minimum $\sum_{1\leq i\leq k}|m_{i}-m/k|$, where $m_{i}=|E(G_{i})|$ and $m=|E(G)|$. We claim that $|m_{i}-m/k|<1$ for every color $c_{i}$. Suppose, to the contrary, that there is a color $c_{i}$ with $|m_{i}-m/k|\geq 1$. We may assume that $m_{i}\geq m/k+1$; as the proof of the case $m_{i}\leq m/k-1$ is similar. Thus there is another color $c_{j}$ with $c_{j}<m/k$ so that $m_{i}>m_{j}+1$. Let $H$ be the factor of $G_{0}$ consisting of the edges that are colored with $c_{i}$ or $c_{j}$. According to the proper edge coloring of $G_{0}$, the graph $H$ must be the union of some paths and cycles. Since all cycles have even size and $m_{i}>m_{j}$, it is easy to check that there is a path $P$ such that whose edges are colored alternatively by $c_{i}$ and $c_{j}$, and also whose end edges are colored with $c_{i}$. Now, it is enough to exchange the colors of the edges of $P$ to find another proper edge-coloring satisfying $m^{\prime}_{i}=m_{i}-1$ and $m^{\prime}_{j}=m_{j}+1$ where $m^{\prime}_{i}$ and $m^{\prime}_{j}$ are the number of edges having these new colors. Since $|m^{\prime}_{j}-m/k|+|m^{\prime}_{i}-m/k|<(|m_{j}-m/k|+1)+(|m_{i}-m/k|-1)$, one can easily derive at a contradiction. Therefore, $G_{1},\ldots,G_{k}$ are the desired factors we are looking for.

Now, we are going to prove item (ii). Let us add some artificial edges to the graph $G_{0}$. For every vertex $u$ with $d_{G}^{+}(u)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(u)\stackrel{{\scriptstyle k}}{{\not\equiv}}0$, denote by $u_{0}^{+}$ and $u_{0}^{-}$ the two vertices in $G_{0}$ having the same degree strictly less than $k$ obtained from splitting of $u^{+}$ and $u^{-}$. Add some new artificial edges between these two vertices to construct these vertices with degree $k$. Apply this method for all such vertices $u$ described above. Consider a proper edge-coloring with at most $k$ colors $c_{1}\ldots,c_{k}$ for the new resulting bipartite graph $G^{\prime}_{0}$. Similarly, we define $G_{i}$ to be the factor of $G$ consisting of the edges of $G$ corresponding to the edges with color $c_{i}$ in $G_{0}$. Note that for every vertex $u$ with $d_{G}^{+}(u)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(u)\stackrel{{\scriptstyle k}}{{\not\equiv}}0$, either one edge between $u_{0}^{+}$ and $u_{0}^{-}$ in $G^{\prime}_{0}$ (may be an artificial edge) is colored with $c_{i}$ or two edges incident with $u_{0}^{+}$ and $u_{0}^{-}$ in $G_{0}$ are colored with $c_{i}$. Thus for every vertex $u$ with $d_{G}^{+}(u)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(u)$, we must have $d^{-}_{G_{i}}(u)-d^{-}_{G}(u)/k=d^{+}_{G_{i}}(u)-d^{+}_{G}(u)/k$ and so $d_{G_{i}}(u)\stackrel{{\scriptstyle 2}}{{\equiv}}d^{+}_{G_{i}}(u)-d^{-}_{G_{i}}(u)=(d_{G}^{+}(u)-d_{G}^{-}(u))/k$. Therefore, $G_{1},\ldots,G_{k}$ are again the desired factors we are looking for.

• $\Box$*

Corollary 2.2

.([2])* Every graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each factor $G_{i}$, $|\,|E(G_{i})|-|E(G)|/k\,|<1$, and for each $v\in V(G_{i})$,*

[TABLE]

**Proof. **

First, consider an orientation for $G$ such that for each vertex $v$, $|d^{+}_{G}(v)-d^{-}_{G}(v)|\leq 1$. Next, apply Theorem 2.1 (i) to this directed graph.

• $\Box$*

3 Factorizations of edge-connected graphs

In this section, we are going to give a sufficient condition for the existence of equitable factorizations in highly edge-connected graphs. For this purpose, let us recall the following lemma from [12, 22].

Lemma 3.1

.([12, 22])* Let $G$ be a graph, let $k$ be an integer, $k\geq 2$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(3k-3)$-edge-connected or $(2k-2)$-tree-connected, then it admits a $p$-orientation modulo $k$. Furthermore, the result holds for loopless $(2k-1)$-edge-connected essentially $(3k-3)$-edge-connected graphs provided that $p(v)=0$ or $p(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}(v)$ for each vertex $v$.*

Before stating the main result, we begin with the following theorem that only needs replacing a weaker condition on a single arbitrary vertex.

Theorem 3.2

.* Let $G$ be a graph with $z\in V(G)$ and let $k$ be a positive integer. If $G$ is $(3k-3)$-edge-connected or $(2k-2)$-tree-connected, then it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ satisfying $||E(G_{i})|-|E(G)|/k|<1$ such that for each $v\in V(G_{i})$,*

[TABLE]

Furthermore, for the vertex $z$, we can also have $|d_{G_{i}}(z)-d_{G_{j}}(z)|\leq 2$.

**Proof. **

According to Lemma 3.1, the graph $G$ admits an orientation such that for each $v\in V(G)\setminus z$, $d^{+}_{G}(v)\stackrel{{\scriptstyle k}}{{\equiv}}0$, and $d^{+}_{G}(z)\stackrel{{\scriptstyle k}}{{\equiv}}|E(G)|$. Now, it is enough to apply Theorem 2.1 (i).

• $\Box$*

Now, we are reedy to state the main result of this section which is a strengthened version of Theorem 1.4.

Theorem 3.3

.* Let $G$ be a graph and let $k$ be a positive integer. Assume that $G$ is $(3k-3)$-edge-connected or $(2k-2)$-tree-connected. If there is a vertex set $Z$ with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$, then $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ satisfying $||E(G_{i})|-|E(G)|/k|<1$ such that for each $v\in V(G_{i})$,*

[TABLE]

Furthermore, the result holds for loopless $(2k-1)$-edge-connected essentially $(3k-3)$-edge-connected graphs.

**Proof. **

For each $v\in Z$, define $p(v)=d_{G}(v)$, and for each $v\in V(G)\setminus Z$, define $p(v)=0$. By the assumption, we have $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Thus by Lemma 3.1, the graph $G$ admits a $p$-orientation modulo $k$ so that for each $v\in Z$, $d^{-}_{G}(v)$ is visible by $k$ and for each $v\in V(G)\setminus Z$, $d^{+}_{G}(v)$ is divisible by $k$. Hence the assertion follows from Theorem 2.1 (i).

• $\Box$*

Graphs with size divisible by $k$ are natural candidates for graphs satisfying the assumptions of Theorem 3.3. We examine them to deduce the following corollary.

Corollary 3.4

.* Let $G$ be a graph of size divisible by $k$. If $G$ is $(3k-3)$-edge-connected or $(2k-2)$-tree-connected, then it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ with the same size such that for each $v\in V(G_{i})$,*

[TABLE]

**Proof. **

Apply Theorem 3.3 with $Z=\emptyset$. Note that $|E(G)|$ is divisible by $k$.

• $\Box$*

The next corollary gives a criterion for the existence of a $3$-equitable factorization in edge-connected graphs.

Corollary 3.5

.* Let $G$ be a $6$-edge-connected graph. Then $G$ has not exactly one vertex $z$ with $d_{G}(z)\stackrel{{\scriptstyle 3}}{{\not\equiv}}0$ if and only if it can be edge-decomposed into three factors $G_{1}$, $G_{2}$, and $G_{3}$ such that for each $v\in V(G_{i})$,*

[TABLE]

Furthermore, the result holds for loopless $5$-edge-connected essentially $6$-edge-connected graphs.

**Proof. **

Set $Z_{0}$ to be the set of all vertices of $G$ such that whose degrees are not divisible by $3$. First assume that $|Z_{0}|\neq 1$. If $|Z_{0}|=0$, then $2|E(G)|$ must be divisible by $3$ and so $|E(G)|$ must be divisible by $3$. Therefore, it is easy to check that there is a subset $Z$ of $Z_{0}$ such that $|E(G)|\stackrel{{\scriptstyle 3}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$ whether $|Z_{0}|\geq 2$ or not. Hence by Theorem 3.3, the graph $G$ has the desired factorization. Now, assume $Z_{0}=\{z\}$. Suppose, to the contrary, that $G$ can be edge-decomposed into three factors $G_{1}$, $G_{2}$, and $G_{3}$ such that for each $v\in V(G_{i})$, $|d_{G_{i}}(v)-d_{G}(v)/3|<1$. We may therefore assume that $d_{G_{1}}(z)=\lfloor d_{G}(z)/3\rfloor$ and $d_{G_{2}}(z)=\lceil d_{G}(z)/3\rceil$. On the other hand, for all vertices $v\in V(G)\setminus\{z\}$, $d_{G_{1}}(v)=d_{G_{2}}(v)=d_{G}(v)/3$ which implies that $d_{G_{1}}(z)$ and $d_{G_{2}}(z)$ have the same parity, because of the handshaking lemma. This is contradiction, as desired.

• $\Box$*

As we observed above, graphs with a number of vertices whose degrees are not divisible by $k$ are other natural candidates for graphs satisfying the assumptions of Theorem 3.3. By employing the following lemma, we examine a special case of them to imply the next corollary.

Lemma 3.6

.(Chowla [6])* Let $k$ be an integer number with $k\geq 2$ and let $x_{1},\ldots,x_{k-1}\in Z_{k}\setminus\{0\}$ be $k-1$ integer numbers coprime with $k$ (not necessarily distinct). If $m\in Z_{k}$, then there is a subset $Z\subseteq\{1,\ldots,k-1\}$ such that $m\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{i\in Z}x_{i}$.*

Corollary 3.7

.* Let $G$ be a $(3k-3)$-edge-connected graph where $k$ is a prime number. If $G$ contains at least $k-1$ vertices whose degree are coprime with $k$, then it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ satisfying $||E(G_{i})|-|E(G)|/k|<1$ such that for each $v\in V(G_{i})$, $|d_{G_{i}}(v)-d_{G}(v)/k|<1.$*

**Proof. **

Let $Z_{0}$ be the set of all vertices of $G$ such that whose degrees are coprime with $k$. Since $|Z_{0}|\geq k-1$, there is a subset $Z$ of $Z_{0}$ such that $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in Z}d_{G}(v)$ according to Lemma 3.6. Hence the assertion follows from Theorem 3.3 immediately.

• $\Box$*

It is perhaps surprising that the lower bound of $k-1$ in the above-mentioned corollary is best possible according to the following observation. It remains to decide whether the statement of Corollary 3.7 holds if $G$ contains at least $k-1$ vertices whose degree are not divisible by $k$. The statement is obviously true for all prime numbers $k$.

Observation 3.8

.* For every integer $k$ with $k\geq 2$, there are infinitely many highly edge-connected graphs $G$ having at least $k-2$ vertices whose degrees are coprime with $k$, while $G$ cannot be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $|d_{G_{i}}(v)-d_{G}(v)/k|<1$.*

**Proof. **

Let $r$ be an arbitrary odd positive integer. Choose a $kr$-edge-connected graph $G$ of odd order in which whose degrees are $rk$ except for $k-2$ vertices having degree $kr+1$ (for $k$ even, one can consider a $kr$-edge-connected $kr$-regular large graph of odd order and insert a new perfect matching of size $k/2-1$ to it). If $G$ has the desired factorization, then according to the vertex degree, there must be a factor $G_{i}$ such that for all vertices $v$, $d_{G_{i}}(v)=\lfloor d_{G}(v)/k\rfloor$. This implies that $G$ an $r$-regular factor which is not possible, because of the handshaking lemma. Hence the assertion holds.

• $\Box$*

4 Parity factorizations of odd-edge-connected graphs

In this section, we are going to develop each of Theorems 1.3 and 5.13 in two ways based on Theorem 2.1. For this purpose, we first need the following lemma which improves the edge-connectivity needed in Lemma 3.1 for a special case.

Lemma 4.1

.([22])* Let $k$ be an odd positive integer. If $G$ is an odd-$(3k-2)$-edge-connected graph, then it admits an orientation such that for each vertex $v$, $d_{G}^{+}(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(v)$.*

The following theorem generalizes Theorem 5.13 to non-Eulerian graphs for the case that $k$ is odd. This result can also be considered as an improvement of a result due to Shu, Zhang, and Zhang (2012) [24] who proved this result for odd-$(2k-1)$-edge-connected graphs without considering the restriction on vertex degrees.

Theorem 4.2

.* Let $k$ be an odd positive integer. If $G$ is an odd-$(3k-2)$-edge-connected or $(2k-2)$-tree-connected graph, then it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}d_{G}(v)$ and*

[TABLE]

**Proof. **

According to Lemmas 3.1 and 4.1, the edge-connectivity condition implies the graph $G$ admits an orientation such that for each vertex $v$, $d_{G}^{+}(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}^{-}(v)$. By Theorem 2.1 (ii), the graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<1$ and $|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|<1$, and also $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}(d_{G}^{+}(v)-d_{G}^{-}(v))/k$. Therefore, $|d_{G_{i}}(v)-d_{G}(v)/k|\leq|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|+|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<2$, and all $d_{G_{i}}(v)$ have the same parity for which $1\leq i\leq k$. Since $d_{G}(v)=\sum_{1\leq j\leq k}d_{G_{j}}(v)$ and $k$ is odd, $d_{G_{i}}(v)$ and $d_{G}(v)$ must have the same parity. Hence the theorem holds.

• $\Box$*

The edge-connectivity needed in Lemma 3.1 can also be improved for the following special case. We are going to apply it to deduce the next result on Eulerian graphs.

Lemma 4.3

.([12])* Let $k$ be an even positive integer, let $G$ be an Eulerian graph, and let $Q\subseteq V(G)$ with $|Q|$ even. If $d_{G}(X)\geq 3k-2$ for every $X\subseteq V(G)$ with $|X\cap Q|$ odd, then $G$ admits an orientation such that for each $v\in V(G)\setminus Q$, $d_{G}^{+}(v)=d_{G}^{-}(v)$, and for each $v\in Q$, $|d_{G}^{+}(v)-d_{G}^{-}(v)|=k$.*

The next theorem strengthens Theorem 5.13 by imposing a new parity restriction on degrees of the desire factors for graphs with higher edge-connectivity.

Theorem 4.4

.* Let $k$ be an even positive integer, let $G$ be an Eulerian graph, and let $f:V(G)\rightarrow Z_{2}$ be a mapping with $\sum_{v\in V(G)}f(v)$ even. If $d_{G}(X)\geq 3k-2$ for every vertex set $X$ with $\sum_{v\in X}f(v)$ odd, or $G$ is $(2k-2)$-tree-connected, then $G$ can be edge-decomposed into $f$-parity factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,*

[TABLE]

**Proof. **

According to Lemmas 3.1 and 4.3, the edge-connectivity condition implies the graph $G$ admits an orientation such that for each vertex $v$, $d_{G}^{+}(v)=d_{G}^{-}(v)$ when $f(v)$ is even, and $|d_{G}^{+}(v)-d_{G}^{-}(v)|=k$ when $f(v)$ is odd. Thus by Theorem 2.1 (ii), the graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<1$ and $|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|<1$, and also $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}(d_{G}^{+}(v)-d_{G}^{-}(v))/k$. Therefore, $|d_{G_{i}}(v)-d_{G}(v)/k|\leq|d^{+}_{G_{i}}(v)-d^{+}_{G}(v)/k|+|d^{-}_{G_{i}}(v)-d^{-}_{G}(v)/k|<2$ and $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}f(v)$. Hence the theorem holds.

• $\Box$*

An attractive application of Theorems 4.2 and 4.4 is given in the following corollary.

Corollary 4.5

.* Let $k$ and $r$ be two positive integers. Let $G$ be a graph whose degrees lie in the set $\{rk,rk+2,...,rk+2k\}$ for which $r|V(G)|$ is even. If $G$ is $(3k-3)$-edge-connected or $(2k-2)$-tree-connected, then it can be edge-decomposed into $k$ $\{r,r+2\}$-factors.*

**Proof. **

If $k$ is odd, then by Theorem 4.2, the graph $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}d_{G}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}r$ and $r-2\leq d_{G}(v)/k-2<d_{G_{i}}(v)<d_{G}(v)/k+2\leq r+4.$ If $k$ is even, then by applying Theorem 4.4 with $f(v)=r$, these factors can similarly be found such that for each $v\in V(G_{i})$, $d_{G_{i}}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}r$ and $r-2<d_{G_{i}}(v)<r+4.$ This completes the proof.

• $\Box$*

4.1 Graphs with degrees divisible by $k$: regular factorizations

In this subsection, we restrict out attention to graphs with degrees divisible by $k$ and derive some results based on the following reformulation of Theorems 4.2 and 4.4 on this family of graphs.

Theorem 4.6

.* Let $k$ be a positive integer and let $G$ be a graph with size and degrees divisible by $k$. Take $Q$ to be the set of all vertices $v$ with $d_{G}(v)/k$ odd. If for every vertex X with $|X\cap Q|$ odd,*

[TABLE]

then $G$ can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$, $d_{G_{i}}(v)=d_{G}(v)/k$.

**Proof. **

To show a a directive proof, we first consider an orientation for $G$ such that out-degree of each vertex is divisible by $k$ using Lemmas 4.1 and 4.3. Next, it is enough to apply Theorem 2.1 (i).

• $\Box$*

The following corollary partially confirms Conjecture 2 in [26] by giving a supplement for Theorem 1.3.

Corollary 4.7

.* Let $r$ be an odd positive integer. If $G$ is a $kr$-regular graph $G$ of even order satisfying $d_{G}(A)\geq 3k-2$, for every vertex set $A$ with $|A|$ odd, then it can be edge-decomposed into $r$-factors.*

**Proof. **

Since $G$ has even order, its size must be divisible by $k$. Note also that for each vertex $v$, $d_{G}(v)/k$ is odd. Thus the assertion follows from Theorem 4.6 immediately,

• $\Box$*

The following corollary gives a supplement for Theorem 3 in [26].

Corollary 4.8

.* Let $r$ be a positive integer and let $r_{1},\ldots,r_{m}$ be $m$ positive integers satisfying $r=r_{1}+\cdots+r_{m}$ and $r_{i}\geq r/k-1\geq 2$ in which $k$ is positive divisor of $r$ with $r/k$ odd. If $G$ is an $r$-regular graph of even order and for every vertex $X$ with $|X|$ odd,*

[TABLE]

then it can be edge-decomposed into factors $G_{1},\ldots,G_{m}$ such that every graph $G_{i}$ is $r_{i}$-regular.

**Proof. **

Apply Corollary 4.7 along with a similar argument stated in the proof of Theorem 3 in [26].

• $\Box$*

The following corollary is a counterpart of Corollary 4.8 and replaces a weaker edge-connectivity condition compared to Theorem 4 in [26]. The proof technique shows a worthwhile application of this kind of odd-edge-connectivity for working with supergraphs.

Corollary 4.9

.* Let $r$ be a positive integer and let $r_{1},\ldots,r_{m}$ be $m$ positive integers satisfying $r=r_{1}+\cdots+r_{m}$ and $r_{i}\geq r/k-1\geq 2$ in which $k$ is positive divisor of $r$ with $r/k$ odd. If $G$ is a graph with $r|V(G)|$ even satisfying $\Delta(G)\leq r$ and for every vertex X with $|X|$ odd,*

[TABLE]

then it can be edge-decomposed into factors $G_{1},\ldots,G_{m}$ satisfying $\Delta(G_{i})\leq r_{i}$ for each $i$ with $1\leq i\leq m$.

**Proof. **

Add some edges to $G$, as long as possible, such that the resulting graph $G^{\prime}$ has maximum degree at most $r$. Since adding loops is possible and $r|V(G)|$ is even, the graph $G^{\prime}$ must be $r$-regular. Obviously, for every vertex set $X$ with $|X|$ odd, we still have $d_{G^{\prime}}(X)\geq 3k-3$. In addition, since $d_{G^{\prime}}(X)$ and $k$ have the same parity, it implies that $d_{G^{\prime}}(X)\geq 3k-2$. Thus by Corollary 4.8, the graph $G^{\prime}$ can be edge-decomposed into factors $G^{\prime}_{1},\ldots,G^{\prime}_{m}$ such that every graph $G^{\prime}_{i}$ is $r_{i}$-regular. Now, it is enough to induce this factorization for $G$ to complete the proof.

• $\Box$*

5 A sufficient edge-connectivity condition for the existence of a parity factor

In 1956 Hoffman made the following theorem on the the existence factors in bipartite graphs whose degrees are close to $\varepsilon$ of the corresponding degrees in the main graph.

Theorem 5.1

.([16])* Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If $G$ is a bipartite graph, then it has a factor $F$ such that for each vertex $v$, $|d_{F}(v)-\varepsilon d_{G}(v)|<1.$*

In 1983 Kano and Saito generlized Theorem 5.1 to general graphs as the next version.

Theorem 5.2

.([18])* Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If $G$ is a graph, then it has a factor $F$ such that for each vertex $v$, $|d_{F}(v)-\varepsilon d_{G}(v)|\leq 1.$*

In 2007 Correa and Goemans [8] formulated the following factorization version of Theorem 5.1 for bipartite graphs. Later, Correa and Matamala (2008) [7] remarked that it is possible to generalize their result to general graphs by replacing Theorem 5.2 in their proof, and Feige and Singh (2008) [10] introduced an interesting alternative proof for this theorem using linear algebraic techniques.

Theorem 5.3

.([7, 8])* Let $\varepsilon_{1},\ldots,\varepsilon_{k}$ be $k$ nonnegative real numbers with $\varepsilon_{1}+\cdots+\varepsilon_{k}=1$. If $G$ is a graph, then it can be edge-decomposed into $k$ factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,*

[TABLE]

In this section, we shall prove the parity versions of Theorems 5.2 and 5.3 which were mentioned in the introduction as Theorems 1.8 and 1.9. Besides them, we also generalize a recent result in [12]. Our proofs are based on the following well-known lemma due to Lovász (1972).

Lemma 5.4

.([21])* Let $G$ be a graph and let $g$ and $f$ be two integer-valued functions on $V(G)$ satisfying $g(v)\leq f(v)$ and $g(v)\stackrel{{\scriptstyle 2}}{{\equiv}}f(v)$ for each $v\in V(G)$. Then $G$ has a parity $(g,f)$-factor if and only if for all disjoint subsets $A$ and $B$ of $V(G)$,*

[TABLE]

where $\omega_{f}(G,A,B)$ denotes the number of components $G[X]$ of $G\setminus(A\cup B)$ satisfying $\sum_{v\in X}f(v)\stackrel{{\scriptstyle 2}}{{\not\equiv}}d_{G}(X,B)$.

5.1 Parity $\varepsilon$-factors

The following theorem provides a parity version for Theorem 5.2 on edge-connected graphs. Note that some edge-connected versions of that theorem were former studied in [3, 9, 17].

Theorem 5.5

.* Let $G$ be a connected graph, let $f:V(G)\rightarrow Z_{2}$ be a mapping with $\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$, and let $\varepsilon$ be a real number with $0<\varepsilon<1$. If for every nonempty proper subset $X$ of $V(G)$,*

[TABLE]

then $G$ has an $f$-parity factor $F$ such that for each vertex $v$,

[TABLE]

Furthermore, for a given arbitrary vertex $z$, we can arbitrary have $d_{F}(z)\geq\varepsilon d_{G}(z)$ or $d_{F}(z)\leq\varepsilon d_{G}(z)$.

**Proof. **

For each vertex $v$, let us define $g_{0}(v)\in\{\lfloor\varepsilon d_{G}(v)\rfloor-1,\lfloor\varepsilon d_{G}(v)\rfloor\}$ and $f_{0}(v)\in\{\lceil\varepsilon d_{G}(v)\rceil,\lceil\varepsilon d_{G}(v)\rceil+1\}$ such that $g_{0}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}f_{0}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}f(v)$. If $g(z)<\varepsilon d_{G}(z)$ and our goal is to impose that $d_{F}(z)\geq\varepsilon d_{G}(z)$, we will replace $g_{0}(z)$ by $g_{0}(z)+2$, and if $f(z)>\varepsilon d_{G}(z)$ and our goal is to impose that $d_{F}(z)\leq\varepsilon d_{G}(z)$, we will replace $f_{0}(z)$ by $f_{0}(z)-2$. Let $A$ and $B$ be two disjoint vertex subsets of $V(G)$ with $A\cup B\neq\emptyset$. By the definition of $g$ and $f$, we must have

[TABLE]

whether $z\in A\cup B$ or not. Take $P$ to be the collection of all vertex sets $X$ such that $G[X]$ is a component of $G\setminus(A\cup B)$ satisfying $\sum_{v\in X}f(v)\stackrel{{\scriptstyle 2}}{{\not\equiv}}d_{G}(X,B)$. It is easy to check that

[TABLE]

Define $P_{c}=\{X\in P:d_{G}(X,A)>0\text{ and }d_{G}(X,B)>0\}$. Obviously,

[TABLE]

Set $P_{a}=\{X\in P:d_{G}(X,A)=0\}$ and $P_{b}=\{X\in P:d_{G}(X,B)=0\}$. According to the definition, for every $X\in P_{a}$, we have

[TABLE]

which implies that $\sum_{v\in X}(d_{G}(v)-f(v))$ is odd. Similarly, for every $X\in P_{b}$, $\sum_{v\in X}f(v)$ must be odd. Thus by the assumption, we must have

[TABLE]

and also

[TABLE]

Therefore,

[TABLE]

According to Relations (1), (2), and (3), one can conclude that

[TABLE]

When both of the sets $A$ and $B$ are empty, the above-mentioned inequality must automatically hold, because $\omega_{f}(G,\emptyset,\emptyset)=0$. Thus the assertion follows from Lemma 5.4.

• $\Box$*

Remark 5.6

. Note that if we had replaced the weaker condition $|d_{F}(z)-\varepsilon d_{G}(z)|\leq 1$ for the vertex $z$, we could impose this condition for another vertex as well. In particular, we could impose this condition for three arbitrary vertices provided that $\varepsilon=1/2$.

The following corollary is an improved version of the main result in [5] by replacing an odd edge-connectivity condition. Note that $r$-regular bipartite graphs are in the class of odd-$r$-edge-connected graphs.

Corollary 5.7

.([4, 5])* Let $r$ and and $r_{0}$ be two odd positive integers with $r\geq r_{0}$. If $G$ is an odd-$\lceil r/r_{0}\rceil$-edge-connected $r$-regular graph, then it contains an $r_{0}$-factor.*

**Proof. **

Apply Theorem 5.5 with $\varepsilon=r_{0}/r$ and $f(v)=r_{0}$ (mod $2$).

• $\Box$*

Corollary 5.8

.([12])* Let $G$ be a graph and let $f:V(G)\rightarrow Z_{2}$ be a mapping with $\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$. If $G$ is $2$-edge-connected, then it has an $f$-parity factor $F$ such that for each vertex $v$,*

[TABLE]

Furthermore, for a given arbitrary vertex $z$, we can arbitrary have $d_{F}(z)\geq d_{G}(z)/2$ or $d_{F}(z)\leq d_{G}(z)/2$.

**Proof. **

Apply Theorem 5.5 with $\varepsilon=1/2$.

• $\Box$*

Corollary 5.9

.* Let $G$ be a graph and let $f:V(G)\rightarrow Z_{2}$ be a mapping with $\sum_{v\in V(G)}f(v)\stackrel{{\scriptstyle 2}}{{\equiv}}0$. If $G$ is $3$-edge-connected, then it has an $f$-parity factor $F$ such that for each vertex $v$,*

[TABLE]

**Proof. **

Apply Theorem 5.5 with $\varepsilon=2/3$.

• $\Box$*

An immediate consequence of the following corollary was appeared in [20] which says that every $2$-edge-connected graph with minimum degree at least three admits a factor whose degrees are positive and even.

Corollary 5.10

.* Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 2/3$. If $G$ is a $2$-edge-connected graph, then it has an even factor $F$ such that for each vertex $v$,*

[TABLE]

The next corollary provides an improved version for a result in [23] due to Lu, Wang, and Lin (2015) who proved that every $2m$-edge-connected graph with minimum degree at least $2m+1$ contains an even factor with minimum degree at least $2m$.

Corollary 5.11

.* Let $G$ be a $2$-edge-connected graph and let $\varepsilon$ be a real number with $2/3\leq\varepsilon<1$. If $G$ is odd-$\lceil 1/(1-\varepsilon)\rceil$-edge-connected, then it has an even factor $F$ such that for each vertex $v$,*

[TABLE]

When the main graph has no odd edge-cuts, one can derive the following simpler version of Corollaries 5.10 and 5.11.

Corollary 5.12

.* Let $\varepsilon$ be a real number with $0\leq\varepsilon\leq 1$. If $G$ is a graph with even degrees, then it admits an even factor $F$ such that for each vertex $v$,*

[TABLE]

Here, we introduce a simple inductive proof for Theorem 8 in [14] based on Corollary 5.12.

Corollary 5.13

.([14])* Every graph $G$ with even degrees can be edge-decomposed into $k$ even factors $G_{1},\ldots,G_{k}$ such that for each vertex $v$, $|d_{G_{i}}(v)-d_{G}(v)/k|<2.$*

**Proof. **

By induction on $k$. We may assume that $k\geq 2$ as the assertion trivially holds when $k=1$. According to Corollary 5.12, the graph can be edge-decomposed into two even factors $G^{\prime}$ and $G_{k}$ such that for each vertex $v$, $|d_{G_{k}}(v)-\frac{1}{k}d_{G}(v)|<2$. By the induction hypothesis, the graph $G^{\prime}$ can be edge-decomposed into $k-1$ even factors $G_{1},\ldots,G_{k-1}$ such that for each vertex $v$, $|d_{G_{i}}(v)-\frac{1}{k-1}d_{G^{\prime}}(v)|<2.$ We claim that these are the desired factors we are looking for. Let $v\in V(G)$. Since $d_{G^{\prime}}(v)\leq\frac{k-1}{k}d_{G}(v)+2-\frac{1}{k}$, we must have $d_{G_{i}}(v)\leq\frac{1}{k-1}(d_{G^{\prime}}(v)-1)+2\leq\frac{1}{k}(d_{G}(v)+1)+2.$ Since $d_{G}(v)$ is even, $(d_{G}(v)+1)/k$ must not be an even integer number, which implies that $d_{G_{i}}(v)\leq\frac{1}{k}d_{G}(v)+2$. Since $d_{G^{\prime}}(v)$ is even, if $\frac{1}{k}d_{G}(v)$ is an even integer number, then $d_{G^{\prime}}(v)\leq\frac{k-1}{k}d_{G}(v)$. Therefore, since $d_{G_{i}}(v)$ is even, one can conclude that $d_{G_{i}}(v)<\frac{1}{k}d_{G}(v)+2$ whether $\frac{1}{k}d_{G}(v)$ is an integer number or not. Similarly, we have $d_{G_{i}}(v)>\frac{1}{k}d_{G}(v)-2$. Hence the assertion holds.

• $\Box$*

By replacing Corollary 5.12 in the proof of Theorem 5.3, one can formulate a parity version of it as the following theorem. It would be interesting to determine the sharp upper bound on vertex degrees to make another generalization for Corollary 5.12.

Theorem 5.14

.* Let $\varepsilon_{1},\ldots,\varepsilon_{k}$ be $k$ nonnegative real numbers with $\varepsilon_{1}+\cdots+\varepsilon_{k}=1$. If $G$ is a graph with even degrees, then it can be edge-decomposed into $k$ even factors $G_{1},\ldots,G_{k}$ such that for each $v\in V(G_{i})$,*

[TABLE]

**Proof. **

Apply Corollary 5.12 along with the same arguments stated in the proof of Theorem 4 in [8].

• $\Box$*