Bipartite partition-connected factors with small degrees

TL;DR

This paper proves that highly connected graphs contain bipartite factors with small degrees and specific connectivity properties, extending understanding of graph decompositions and their degree constraints.

Contribution

It introduces new results on bipartite $m$-partition-connected factors with bounded degrees in highly connected graphs.

Findings

Existence of bipartite $m$-partition-connected factors with degree at most 75% of original degrees

Bipartite $m$-partition-connected factors with maximum degree at most 3m+1 in tough graphs

Presence of bipartite connected factors with degrees in {k, 2k, 3k, 4k} for large enough graphs

Abstract

In this paper, we show that every $2m$-partition-connected graph $G$ has a bipartite $m$-partition-connected factor $H$ such that for each vertex $v$, $d_H(v)\le \lceil \frac{3}{4}d_G(v)\rceil$. A graph $H$ is said to be $m$-partition-connected, if it contains $m$ edge-disjoint spanning trees. As an application, we conclude that tough enough graphs with appropriate number of vertices have a bipartite $m$-partition-connected factor with maximum degree at most $3m+1$. Finally, we prove that tough enough graphs of order at least $3k$ admit a bipartite connected factor whose degrees lie in the set $\{k,2k,3k,4k\}$.

Equations

$$d_H(X)-d_{H_i}(X)=|E(H)|-|E(H_i)|\ge 0.$$

$$d_{G}(X)=\sum_{1\leq i\leq c}\big{(}d_{G}(X)-d_{H_{i}}(X)\big{)}\geq c\big{(}d_{G}(X)-d_{H}(X)\big{)}.$$

$$\frac{1}{2}\sum_{X\in P}d_{H}(X)\geq\frac{c-1}{c}\sum_{X\in P}\frac{1}{2}d_{G}(X)=\frac{c-1}{c}e_{G}(P)\geq\frac{c-1}{c}\big{(}\sum_{X\in P}\frac{c}{c-1}l(X)\;-\frac{c}{c-1}l(G)\big{)},$$

$$e_H(P)=\frac{1}{2}\sum _{X\in P}{d}_H(X)\ge \sum _{X\in P}l(X)-l(G).$$

$$\text{$\Theta$}_{l}(G_{0}\setminus S)<1+\sum_{v\in S}\big{(}\eta(v)-2l(v)\big{)}+l(G_{0})+l(S)-\lambda(e_{G}(S)+l(S)),$$

$$\text{$\Theta$}_{l}(G_{0}\setminus S)\leq\sum_{v\in S}\big{(}\frac{1}{2}d_{G_{0}}(v)+\frac{\text{$\varepsilon$}}{2}d_{G}(v)-l(v)\big{)}+l(G)-e_{G_{0}}(S).$$

$$\sum _{A\in P}l(A)+\sum _{v\in S}l(v)-l(G)\le \text{ε}{e}_G(P\cup \{\{v\}:v\in S\})=\sum _{A\in P}\frac{\text{ε}}{2}{d}_G(A)+\sum _{v\in S}\frac{\text{ε}}{2}{d}_G(v).$$

$$\text{$\Theta$}_{l}(G_{0}\setminus S)=\sum_{A\in P}l(A)-e_{G_{0}}(P)\leq\sum_{v\in S}\big{(}\frac{1}{2}d_{G_{0}}(v)+\frac{\text{$\varepsilon$}}{2}d_{G}(v)-l(v)\big{)}+l(G)-e_{G_{0}}(S).$$

$$d_H(v)\le \lceil \frac{1}{2k}{d}_{G_0}(v)+\frac{\text{ε}}{2k}{d}_G(v)+\frac{k-1}{k}l(v)\rceil +\max \{0,d_F(v)-l(v)\}.$$

$$\text{$\Theta$}_{kl}(G_{0}\setminus S)\leq\sum_{v\in S}\big{(}\frac{1}{2}d_{G_{0}}(v)+\frac{\text{$\varepsilon$}}{2}d_{G}(v)-kl(v)\big{)}+kl(G)-e_{G_{0}}(S),$$

$$\text{$\Theta$}_{l}(G_{0}\setminus S)\leq\frac{1}{k}\text{$\Theta$}_{kl}(G_{0}\setminus S)<1+\sum_{v\in S}\big{(}\eta(v)-2l(v)\big{)}+l(G)+\frac{k-1}{k}l(S)-\frac{1}{k}e_{G_{0}}(S),$$

$$d_H(v)\le \lceil \frac{1}{2}{d}_{G_0}(v)+\frac{\text{ε}}{2}{d}_G(v)\rceil .$$

$$d_{H}(v)\leq\big{\lceil}\frac{2c-1}{2c}d_{G}(v)\big{\rceil}.$$

$$d_H(v)\le \lceil \frac{1}{2}{d}_{G_0}(v)+\frac{c-1}{2c}{d}_G(v)\rceil \le \lceil \frac{2c-1}{2c}{d}_G(v)\rceil .$$

$$d_H(v)\le \lceil \frac{3}{4}{d}_G(v)\rceil .$$

$$\text{$\Theta$}_{l}(G_{0}\setminus S)\leq\frac{1}{k}\text{$\Theta$}_{kl}(G\setminus S)+\sum_{v\in S}\big{(}\frac{1}{2}d_{G_{0}}(v)-\frac{1}{2k}d_{G}(v)\big{)}-e_{G_{0}}(S)+\frac{1}{k}e_{G}(S).$$

$$\frac{1}{k}{e}_G(P)+\sum _{v\in S}\frac{1}{2k}{d}_G(v)-\frac{1}{k}{e}_G(S)=\sum _{A\in P}\frac{1}{2k}{d}_G(A)\le \sum _{A\in P}\frac{1}{2}{d}_{G_0}(A)=e_{G_0}(P)+\sum _{v\in S}\frac{1}{2}{d}_{G_0}(v)-e_{G_0}(S).$$

$$\text{$\Theta$}_{l}(G_{0}\setminus S)=\sum_{A\in P}l(A)-e_{G_{0}}(P)\leq\frac{1}{k}(\sum_{A\in P}kl(A)\,-e_{G}(P))+\sum_{v\in S}\big{(}\frac{1}{2}d_{G_{0}}(v)-\frac{1}{2k}d_{G}(v)\big{)}-e_{G_{0}}(S)+\frac{1}{k}e_{G}(S).$$

$$0<d_F(v)\le d_H(v)-k+1\le 6k-2-k+1<5k.$$

Full text

*Keywords:

Toughness; bipartite factor; chromatic number; connected factor; partition-connected; modulo factor.*

1 Introduction

In this article, all graphs have no loop, but multiple edges are allowed. Let $G$ be a graph. The vertex set and the edge set of $G$ are denoted by $V(G)$ and $E(G)$, respectively. The degree $d_{G}(v)$ of a vertex $v$ is the number of edges of $G$ incident to $v$. Let $A$ and $B$ be two subsets of $V(G)$. This pair is said to be intersecting, if $A\cap B\neq\emptyset$. Let $l$ be a real function on subsets of $V(G)$ with $l(\emptyset)=0$. For notational simplicity, we write $l(G)$ for $l(V(G))$ and write $l(v)$ for $l(\{v\})$. The function $l$ is said to be supermodular, if for all vertex sets $A$ and $B$, $l(A\cap B)+l(A\cup B)\geq l(A)+l(B).$ Likewise, $l$ is said to be intersecting supermodular, if for all intersecting pairs $A$ and $B$ the above-mentioned inequality holds. The set function $l$ is called nonincreasing, if $l(A)\geq l(B)$, for all nonempty vertex sets $A$, $B$ with $A\subseteq B$. Note that several results of this paper can be hold for real functions $l$ such that $\sum_{v\in A}l(v)-l(A)$ is integer for every vertex set $A$. For clarity of presentation, we will assume that $l$ is integer-valued. The graph $G$ is said to be $l$-edge-connected, if for all nonempty proper vertex sets $A$, $d_{G}(A)\geq l(A)$, where $d_{G}(A)$ denotes the number of edges of $G$ with exactly one end in $A$. Likewise, the graph $G$ is called $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_{G}(P)\geq\sum_{A\in P}l(A)-l(V(G)),$ where $e_{G}(P)$ denotes the number of edges of $G$ joining different parts of $P$. It is known that if $l$ is intersecting supermodular, then the vertex set of $G$ can be expressed uniquely as a disjoint union of vertex sets of some induced $l$-partition-connected subgraphs. These subgraphs are called the $l$-partition-connected components of $G$. An $l$-partition-connected graph $G$ is called minimally $l$-partition-connected, if for every edge $e$ of $G$, the resulting graph $G-e$ is not $l$-partition-connected. To measure $l$-partition-connectivity of $G$, we define the parameter \text{\Theta}_{l}(G)=\sum_{A\in P}l(A)-e_{G}(P), where $P$ is the partition of $V(G)$ obtained from $l$-partition-connected components of $G$. In  [7], it was proved that \text{\Theta}_{l}(G)\leq\frac{1}{k}\text{\Theta}_{kl}(G), when $k$ is a real number with $k\geq 1$. A graph is said to be $m$-tree-connected, it has $m$ edge-disjoint spanning trees. By the main result in [8, 11], a graph $G$ is $m$-tree-connected if and only if it is $m$-partition-connected. Let $t$ be a positive real number, a graph $G$ is said to be $t$-tough, if $\omega(G\setminus S)\leq\max\{1,\frac{1}{t}|S|\}$ for all $S\subseteq V(G)$. The chromatic number of a graph $G$ is to the minimum number of colors needed to color the vertices of $G$ such that adjacent vertices admit different colors. This number is denoted by $\chi(G)$. For a set of integers $A$, an $A$-factor is a spanning subgraph with vertex degrees in $A$. A modulo $k$-regular factor is a spanning subgraph whose degrees are positive and divisible by $k$. Throughout this article, all variables $k$ and $c$ are positive and integer, unless otherwise stated, and all variables $m$ are nonnegative and integer.

In 1973 Chvátal [1] conjectured that there exists a positive real number $t_{0}$ such that every $t_{0}$-tough graph of order at least three admits a Hamiltonian cycle. In 1989 Win [12] confirmed a weaker version of this conjecture as the following result. In 2000 Ellingham and Zha [2] refined this result for graphs with higher toughness by proving that every $4$-tough graph of order at least three admits a connected $\{2,3\}$-factor.

Theorem 1.1

.([12])* Every $1$-tough graph admits a connected factor with maximum degree at most $3$.*

Recently, the present author generalized Theorem 1.1 to following tree-connected version for tough enough graphs. As an application, we concluded that every $4$-tough graph of order at least three admits a connected $\{2,4\}$-factor.

Theorem 1.2

.([5])* Every $m^{2}$-tough graph of order at least $2m$ admits an $m$-tree-connected factor with maximum degree at most $2m+1$.*

In this paper, we turn our attention to investigate bounded degree tree-connected bipartite factors by proving that every $2m$-tree-connected graph $G$ has a bipartite $m$-tree-connected factor $H$ such that for each vertex $v$, $d_{H}(v)\leq\lceil\frac{3}{4}d_{G}(v)\rceil$. From this result, one can form the following bipartite version for Theorem 1.2. As an application, we show that every $16$-tough graph of order at least six admit a bipartite connected $\{2,4,6\}$-factor.

Theorem 1.3

.* Every $4m^{2}$-tough graph of order at least $4m$ admits a bipartite $m$-tree-connected graph with maximum degree at most $3m+1$.*

In 1985 Enomoto, Jackson, Katerinis, and Saito [3] proved that every $k$-tough graph $G$ with $k|V(G)|$ even admits a $k$-factor. In tis paper, we show that tough enough graphs of order at least $3k$ admit a connected bipartite factor whose degrees lie in the set $\{k,2k,3k,4k\}$. More precisely, we apply a combination of the following theorem and Theorem 1.3.

Theorem 1.4

.([6])* Every $(2k-1)$-tree-connected bipartite graph $G$ with $u\in V(G)$ has a connected modulo $k$-regular factor $H$ such that for each $v\in V(H)\setminus u$, $d_{H}(v)\leq d_{G}(v)-k+1$.*

2 Partition-connected factors with small chromatic number

In 2008 Thomassen [9] showed that every $(2\lambda-1)$-edge-connected graph has a bipartite $\lambda$-edge-connected factor. Recently, the present author refined this result to the following tree-connected version.

Theorem 2.1

.([6])* Every $2m$-tree-connected graph $G$ has a bipartite $m$-tree-connected factor $H$ such that for every vertex set $X$, $d_{H}(X)\geq d_{G}(X)/2$.*

In the following theorem, we provide a generalization for Theorem 2.1 by giving a sufficient partition-connected condition for the existence of partition-connected factors with bounded chromatic number.

Theorem 2.2

.* Let $G$ be a graph, let $c$ be a positive integer with $c\geq 2$, and let $l$ be a real function on subsets of $V(G)$. If $G$ $c\,l/(c-1)$-partition-connected, then it has a $c$-partite $l$-partition-connected factor $H$ such that for every vertex set $X$, $d_{H}(X)\geq\frac{c-1}{c}d_{G}(X).$*

**Proof. **

Let $H$ be an induced $c$-partite factor of $G$ with the maximum $|E(H)|$. Let $X$ be a vertex subset of $V(G)$ and let $V_{1},\ldots,V_{c}$ be the partite sets of $H$. For every $i$ with $1\leq i\leq c$, let $X_{i}=X\cap V_{i}$ and define $H_{i}$ to be the induced $c$-partite factor of $G$ with the partite sets $V_{1}^{i},\ldots,V_{c}^{i}$, where $V_{j}^{i}=(V_{j}\setminus X_{j})\cup X_{i+j}$ and $i+j$ is computed modulo $c$. Since $|E(H)|\geq|E(H_{i})|$, we must have

[TABLE]

Therefore,

[TABLE]

This implies that $d_{H}(X)\geq\frac{c-1}{c}d_{G}(X)$. Let $P$ be a partition of $V(G)$. Since $G$ is $cl/(c-1)$-partition-connected, we must have

[TABLE]

which implies that

[TABLE]

Hence $H$ is $l$-partition-connected, as desired.

• $\Box$*

Corollary 2.3

.* Every graph $G$ contains a factor $H$ such that for every vertex set $X$, $d_{H}(X)\geq\frac{c-1}{c}d_{G}(X)$, and $\chi(H)\leq c$, where $c$ is a given arbitrary integer number with $c\geq 2$.*

Corollary 2.4

.* Let $G$ be a graph and let $l$ be a real function on subsets of $V(G)$. If $G$ is $2l$-partition-connected, then it has a bipartite $l$-partition-connected factor $H$ such that for every vertex set $X$, $d_{H}(X)\geq d_{G}(X)/2$.*

3 Partition-connected factors with small chromatic numbers and degrees

In this section, we shall give a sufficient condition for a graph to satisfy the assumption of the following theorem. As an application, we derive a result on partition-connected factors with small chromatic numbers and degrees.

Theorem 3.1

.([7])* Let $G_{0}$ be a graph with the factor $F$ and let $l$ be an intersecting supermodular nonincreasing nonnegative integer-valued function on subsets of $V(G_{0})$. Let $\lambda\in[0,1]$ be a real number and let $\eta$ be a real function on $V(G_{0})$. If for all $S\subseteq V(G_{0})$,*

[TABLE]

then $G_{0}$ has an $l$-partition-connected factor $H$ containing $F$ such that for each vertex $v$, $d_{H}(v)\leq\lceil\eta(v)-\lambda l(v)\rceil+\max\{0,d_{F}(v)-l(v)\}.$

Before stating the main result, let us make the following lemma.

Lemma 3.2

.* Let $G$ be a graph with the spanning subgraph $G_{0}$, let $l$ be an intersecting supermodular real function on subsets of $V(G)$, and let $\varepsilon$ be a positive real number. If $G$ is l/\text{\varepsilon}-partition-connected and d_{G_{0}}(X)\geq\text{\varepsilon}\,d_{G}(X) for every $X\subseteq V(G)$, then for every $S\subseteq V(G)$,*

[TABLE]

**Proof. **

Let $P$ be a partition of $V(G)\setminus S$ obtained from the $l$-partition-connected components of $G\setminus S$. By the assumption, \sum_{A\in P}\frac{\text{\varepsilon}}{2}d_{G}(A)\leq\sum_{A\in P}\frac{1}{2}d_{G_{0}}(A)=e_{G_{0}}(P)+\sum_{v\in S}\frac{1}{2}d_{G_{0}}(v)-e_{G_{0}}(S). Since $G$ is l/\text{\varepsilon}-partition-connected, we must have

[TABLE]

Therefore,

[TABLE]

Hence the lemma is proved.

• $\Box$*

Now, we are ready to prove the main result of this section.

Theorem 3.3

.* Let $G$ be a graph, let $l$ be an intersecting supermodular nonincreasing nonnegative integer-valued function on subsets of $V(G)$, and let $k$ and $\varepsilon$ be two positive real numbers with k\geq 1\geq\text{\varepsilon}. Let $G_{0}$ be a spanning subgraph of $G$ with a factor $F$. If $G$ is kl/\text{\varepsilon}-partition-connected and d_{G_{0}}(X)\geq\text{\varepsilon}\,d_{G}(X) for every $X\subseteq V(G)$, then $G_{0}$ has an $l$-partition-connected factor $H$ containing $F$ such that for each vertex $v$,*

[TABLE]

Furthermore, for a given arbitrary vertex $u$, the upper bound can be reduced to \lfloor\frac{1}{2k}d_{G_{0}}(u)+\frac{\text{\varepsilon}}{2k}d_{G}(u)+\frac{k-1}{k}(l(u)-l(G))\rfloor+\max\{0,d_{F}(u)-l(u)\}.

**Proof. **

Let $S\subseteq V(G)$. By Lemma 3.2,

[TABLE]

which implies that

[TABLE]

where \eta(u)=\frac{1}{2k}d_{G_{0}}(u)+\frac{\text{\varepsilon}}{2k}d_{G}(u)+l(u)-\frac{k-1}{k}l(G)-q and $q$ is a real number with $0\leq q<1$ which is sufficiently close to $1$, and \eta(v)=\frac{1}{2k}d_{G_{0}}(v)+\frac{\text{\varepsilon}}{2k}d_{G}(v)+l(v) for all vertices $v$ with $v\neq u$. Now, it is enough to apply Theorem 3.1 with $\lambda=1/k$.

• $\Box$*

For the special case $k=1$, the theorem becomes simpler as the following corollary.

Corollary 3.4

.* Let $G$ be a graph with the spanning subgraph $G_{0}$, let $l$ be an intersecting supermodular nonincreasing nonnegative integer-valued function on subsets of $V(G)$, and let $\varepsilon$ be a positive real number. If $G$ is l/\text{\varepsilon}-partition-connected and d_{G_{0}}(X)\geq\text{\varepsilon}\,d_{G}(X) for every $X\subseteq V(G)$, then $G_{0}$ has an $l$-partition-connected factor $H$ such that for each vertex $v$,*

[TABLE]

Furthermore, for a given arbitrary vertex $u$, the upper bound can be reduced to \lfloor d_{G_{0}}(u)/2+\text{\varepsilon}\,d_{G}(u)/2\rfloor.

An application of Corollary 3.4 and Theorem 2.2 is given in the next result.

Corollary 3.5

.* Every $mc/(c-1)$-partition-connected graph $G$ with $c\geq 2$ has a $c$-partite $m$-tree-connected factor $H$ such that for each vertex $v$,*

[TABLE]

Furthermore, for an arbitrary given vertex $u$, the upper bound can be reduced to $\lfloor\frac{2c-1}{2c}d_{G}(u)\rfloor$.

**Proof. **

By Theorem 2.2, the graph $G$ has a $c$-partite $m$-tree-connected factor $G_{0}$ such that for every vertex set $X$, $d_{G_{0}}(X)\geq(c-1)d_{G}(X)/c$. By applying Theorem 3.4 with \text{\varepsilon}=(c-1)/c, the graph $G_{0}$ has an $m$-tree-connected factor $H$ such that for each vertex $v$,

[TABLE]

Hence the proof is completed.

• $\Box$*

Corollary 3.6

.* Every $2m$-tree-connected graph $G$ has a bipartite $m$-tree-connected factor $H$ such that for each vertex $v$,*

[TABLE]

Furthermore, for an arbitrary given vertex $u$, the upper bound can be reduced to $\lfloor\frac{3}{4}d_{G}(u)\rfloor.$

The next theorem gives another sufficient condition for a graph to satisfy the assumption of Theorem 3.1.

Theorem 3.7

.* Let $G$ be a graph with the spanning subgraph $G_{0}$, let $k$ be a real number with $k\geq 1$, and let $l$ be an intersecting supermodular real function on subsets of $V(G)$. Assume that for every vertex set $A$, $d_{G_{0}}(A)\geq\frac{1}{k}d_{G}(A)$. If $S\subseteq V(G)$, then*

[TABLE]

**Proof. **

Let $P$ be a partition of $V(G)\setminus S$ obtained from the $l$-partition-connected components of $G\setminus S$. By the assumption,

[TABLE]

Therefore,

[TABLE]

This inequality can complete the proof.

• $\Box$*

4 Bipartite connected modulo regular factors with small degrees

In 2014 Thomassen [10] showed that graphs with high enough edge-connectivity admit a bipartite modulo $k$-regular factor. This result is recently developed to the following connected factor version.

Theorem 4.1

.([6])* Every $(4k-2)$-tree-connected graph admits a bipartite connected modulo $k$-regular factor.*

In the following, we shall give a sufficient toughness condition for the existence of a bipartite connected modulo regular factor with small degrees.

Theorem 4.2

.* Every $16k^{2}$-tough graph of order at least $3k$ admits a bipartite connected $\{k,2k,3k,4k\}$-factor.*

**Proof. **

If $|V(G)|<8k-4$, the graph $G$ must be complete and the proof is straightforward. So, suppose $|V(G)|\geq 8k-4$. By Theorem 1.3, the graph $G$ has a bipartite $(2k-1)$-tree-connected factor $H$ such that $\Delta(H)\leq 3(2k-1)+1=6k-2$. We may asume that $H$ is minimally $(2k-1)$-tree-connected and so it has a vertex $u$ with $d_{H}(u)\leq 2(2k-1)$. By Theorem 1.4, the graph $H$ has a connected factor $F$ with degrees divisible by $k$ such that for each $v\in V(F)\setminus u$,

[TABLE]

In addition, for the special vertex $u$, we must automatically have $d_{F}(u)\leq 4k$. Hence $F$ is the desired factor we are looking for.

• $\Box$*

A special case of Theorem 4.2 can be refined to the following stronger version.

Theorem 4.3

.* Every $16$-tough graph of order at least six admits a bipartite connected $\{2,4,6\}$-factor.*

**Proof. **

If $|V(G)|<16$, the graph $G$ must be complete and the proof is straightforward. If $|V(G)|\geq 16$, then by Theorem 1.3, the graph $G$ has a bipartite $2$-tree-connected factor $H$ such that $\Delta(H)\leq 7$. Now, it is enough to consider a spanning Eulerian subgraph of $H$ [4].

• $\Box$*