Binding Number and Fractional k-Factors of Graphs

Abstract

In this paper, we consider the relationship between the binding number and the existence of fractional k-factors of graphs. The binding number of G is defined by Woodall as $bind\left(G\right)=\mathrm{min}\left\{\frac{|{N}_{G}\left(X\right)|}{|X|}:\varnothing \ne X\subseteq V\left(G\right)\right\}$ . It is proved that a graph G has a fractional 1-factor if $bind\left(G\right)\ge 1$ and has a fractional k-factor if $bind\left(G\right)\ge k-\frac{1}{k}$ . Furthermore, it is showed that both results are best possible in some sense.

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Chang, R. (2024) Binding Number and Fractional k-Factors of Graphs. Journal of Applied Mathematics and Physics, 12, 2594-2600. doi: 10.4236/jamp.2024.127154.

1. Introduction

All graphs considered in this paper are assumed to be infinite and simple. We refer the readers to [1] for the terminologies not defined here. Let G be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$ . For $x\in V\left(G\right)$ , the degree of x in G is denoted by ${d}_{G}\left(x\right)$ . The minimum vertex degree of G is denoted by $\delta \left(G\right)$ . For any $S\subseteq V\left(G\right)$ , we denote by ${N}_{G}\left(S\right)$ the neighborhood set of S in G and we use $G\left[S\right]$ and $G-S$ to denote the subgraph of G induced by S and $V\left(G\right)-S$ , respectively. A subset S of $V\left(G\right)$ is called an independent set (a covering set) of G if every edge of G is incident with at most (at least) one vertex of S.

Let g and f be two integer-valued functions defined on $V\left(G\right)$ with $g\left(x\right)\le f\left(x\right)$ for any $x\in V\left(G\right)$ . A spanning subgraph F of G is called a $\left(g,f\right)$ -factor if $g\left(x\right)\le {d}_{F}\left(x\right)\le f\left(x\right)$ holds for any vertex $x\in V\left(G\right)$ . A $\left(g,f\right)$ -factor is called an $\left[a,b\right]$ -factor if $g\left(x\right)=a$ and $f\left(x\right)=b$ for any $x\in V\left(G\right)$ . An $\left[a,b\right]$ -factor is called a k-factor if $a=b=k$ .

Let $h:E\left(G\right)\to \left[0,1\right]$ be a function, and let $k\ge 1$ be an integer. If ${\sum }_{e\ni x}h\left(e\right)=k$ holds for each vertex $x\in V\left(G\right)$ , we call $G\left[{F}_{h}\right]$ a fractionalk-factor of G with indicator function h where ${F}_{h}=\left\{e\in E\left(G\right)|h\left(e\right)>0\right\}$ . A fractional 1-factor is also called a fractional perfect matching [2].

The binding number of G is defined by Woodall [3] as

$bind\left(G\right)=\mathrm{min}\left\{\frac{|{N}_{G}\left(X\right)|}{|X|}:\varnothing \ne X\subseteq V\left(G\right)\right\}$ . It is trivial by the definition that $bind\left(G\right)\ge c$ implies that for every subset $X\subseteq V\left(G\right)$ , we have either ${N}_{G}\left(X\right)=V\left(G\right)$ or $|{N}_{G}\left(X\right)|\ge c|X|$ . It is also obvious that if $bind\left(G\right)>1$ , then G is connected. Many authors have investigated the relationship between binding number and the existence of factors in graphs. For more information, please refer to [4]-[6]. We begin with some known results.

Anderson gave a sufficient condition for the existence of 1-factors.

Theorem 1.1. ([7]) If a graph G has even order and $bind\left(G\right)\ge \frac{4}{3}$ , then G has a 1-factor.

Woodall showed the relationship of the binding number and the existence of a Hamiltonian cycle in a graph. In [8] obtained the binding number condition for restricted matching extension in graphs.

Katerinis and Woodal [9] and Katerinis [10] found the minimum degree and binding number conditions for a graph to have k-factors. Zhou obtained the binding number condition for a graph to be ID-k-factor-critical [11].

[12] considered binding number and the existence of f-factors in graphs. The researchers discussed binding number and the existence of $\left(g,f\right)$ -factors in graphs [13] and binding number and the existence of $\left(g,f\right)$ -factors with prescribed properties in graphs [14]. The authors studied binding number and the existence of fractional $\left(g,f\right)$ -factor-critical in graphs [15]. Recently, the researchers considered binding number conditions and various factors ([16]-[18]).

In this paper, we consider the relationship between the binding number and the existence of fractional k-factors in G. Our main results are the following two theorems.

Theorem 1.2. Let G be a connected graph with $|V\left(G\right)|\ge 2$ , then G has a fractional perfect matching if $bind\left(G\right)\ge 1$ .

Theorem 1.3. Let $k\ge 2$ be an integer. A Graph G with $|V\left(G\right)|\ge k+1$ has a fractional k-factor if $bind\left(G\right)\ge k-\frac{1}{k}$ .

2. Preliminary Lemmas

Liu and Zhang gave a necessary and sufficient condition for a graph to have a fractional k-factor in [19].

Lemma 2.1. ([19]) Let $k\ge 1$ be an integer. A graph G has a fractional k-factor if and only if for any subset S of $V\left(G\right)$ ,

$k|T|-{d}_{G-S}\left(T\right)\le k|S|$

where $T=\left\{x\in V\left(G\right)-S|{d}_{G-S}\left(x\right)\le k-1\right\}$ .

In particular, for $k=1$ , Scheinermman obtained the following result.

Lemma 2.2. A graph G has a fractional perfect matching if and only if for any subset S of $V\left(G\right)$ ,

$i\left(G-S\right)\le |S|$

where $i\left(G-S\right)=\left\{x\in V\left(G\right)-S|{d}_{G-S}\left(x\right)=0\right\}$ .

We obtained the following result in [20].

Lemma 2.3. ([20]) Let G be a graph and $H=G\left[T\right]$ such that ${d}_{G}\left(x\right)=k-1$ for every $x\in V\left(H\right)$ and no component of H is isomorphic to ${K}_{k}$ where $T\subseteq V\left(G\right)$ and $k\ge 2$ . Then H has a maximal independent set I and a covering set $C=V\left(G\right)-I$ satisfying

$|V\left(H\right)|\le \left(k-\frac{1}{k+1}\right){i}^{\prime }+\sum _{j=0}^{k-2}\left(j+1\right){{i}^{″}}_{j},\text{\hspace{0.17em}}\text{\hspace{0.17em}}|C|\le \left(k-1-\frac{1}{k+1}\right){i}^{\prime }+\sum _{j=0}^{k-2}\text{ }\text{ }j{{i}^{″}}_{j},$

where ${i}^{\prime }=|{I}^{\prime }|=|\left\{x|x\in I,{d}_{H}\left(x\right)=k-1={d}_{G}\left(x\right)\right\}|$ , ${{i}^{″}}_{j}=|\left\{x|x\in {I}^{″}=I-{I}^{\prime },{d}_{H}\left(x\right)=j<{d}_{G}\left(x\right)\right\}|$ .

Lemma 2.4. ([21]) Let G be a graph and $H=G\left[T\right]$ such that $\delta \left(H\right)\ge 1$ and $1\le {d}_{G}\left(x\right)\le k-1$ for every $x\in V\left(H\right)$ where $T\subseteq V\left(G\right)$ and $k\ge 2$ . Let ${T}_{1},\cdots ,{T}_{k-1}$ be a partition of the vertices of H satisfying ${d}_{G}\left(x\right)=j$ for each $x\in {T}_{j}$ where we allow some ${T}_{j}$ to be empty. If each component of H has a vertex of degree at most $k-2$ in G, then H has a maximal independent set I and a covering set $C=V\left(H\right)-I$ such that

$\sum _{j=1}^{k-1}\left(k-j\right){c}_{j}\le \sum _{j=1}^{k-1}\left(k-2\right)\left(k-j\right){i}_{j}$

where ${c}_{j}=|C\cap {T}_{j}|$ and ${i}_{j}=|I\cap {T}_{j}|$ for $j=1,\cdots ,k-1$ .

3. Proof of Theorems

Suppose that G satisfies the conditions in Theorem 1.2, but G has no fractional perfect matching. By Lemma 2.2, there exists a subset S of $V\left(G\right)$ such that $i\left(G-S\right)>|S|$ . We choose X as the set of isolated vertices of $G-S$ , that is, $|X|=i\left(G-S\right)$ . Since G is connected, it follows that $S\ne \varnothing$ , and $|{N}_{G}\left(X\right)|\subseteq S\ne V\left(G\right)$ .

According to the definition of $bind\left(G\right)$ , we obtain $bind\left(G\right)\le \frac{|{N}_{G}\left(X\right)|}{|X|}\le \frac{|S|}{i\left(G-S\right)}<1$ , contradicting with $bind\left(G\right)\ge 1$ . $\square$

Proof of Theorem 1.3. Suppose that G satisfies the conditions in Theorem 1.3, but G has no fractional k-factors. From Lemma 2.1, there exists a subset S of $V\left(G\right)$ such that

$k|T|-{d}_{G-S}\left(T\right)>k|S|,$ (1)

where $T=\left\{x\in V\left(G\right)-S|{d}_{G-S}\left(x\right)\le k-1\right\}$ .

Let m be the number of the components of ${H}^{\prime }=G\left[T\right]$ which are isomorphic to ${K}_{k}$ , we may assume that ${C}_{1},{C}_{2},\cdots ,{C}_{m}$ are these components. Let ${T}_{0}=\left\{x\in V\left({H}^{\prime }\right)|{d}_{G-S}\left(x\right)=0\right\}$ and $H={H}^{\prime }-m{K}_{k}-{T}_{0}$ .

If $|V\left(H\right)|=0$ , by (1) we get $k|{T}_{0}|+mk>k|S|$ , that is, $|S|\le |{T}_{0}|+m-1$ .

If $m=0$ , $|S|<|{T}_{0}|$ . Set $X={T}_{0}$ , then $|X|=|{T}_{0}|$ and ${N}_{G}\left(X\right)\subseteq S$ , ${N}_{G}\left(X\right)\ne V\left(G\right)$ . We obtain that $bind\left(G\right)\le \frac{|{N}_{G}\left(X\right)|}{|X|}\le \frac{|S|}{|{T}_{0}|}<1$ , a contradiction.

If $m\ge 1$ , let $X={C}_{1}\cup \cdots \cup {C}_{m-1}\cup \left\{x\right\}\cup {T}_{0}$ where x is an arbitrary vertex of ${C}_{m}$ , then $|X|=\left(m-1\right)k+1+|{T}_{0}|$ and ${N}_{G}\left(X\right)\subseteq {C}_{1}\cup \cdots \cup {C}_{m-1}\cup \left({C}_{m}-\left\{x\right\}\right)\cup S$ , ${N}_{G}\left(X\right)\ne V\left(G\right)$ . This follows that

$bind\left(G\right)\le \frac{|{N}_{G}\left(X\right)|}{|X|}\le \frac{mk-1+|S|}{\left(m-1\right)k+1+|{T}_{0}|}\le \frac{mk-1+m-1+|{T}_{0}|}{\left(m-1\right)k+1+|{T}_{0}|}.$

When $k\ge 2$ ,

$\begin{array}{l}\left(\left(m-1\right)k+1+|{T}_{0}|\right)\left(k-\frac{1}{k}\right)-\left(mk-1+m-1+|{T}_{0}|\right)\\ \ge \left(\left(m-1\right)k+1\right)\left(k-\frac{1}{k}\right)-\left(mk-1+m-1\right)=\left(m-1\right)\left({k}^{2}-k-2\right)+1-\frac{1}{k}>0.\end{array}$

That is, $bind\left(G\right) , a contradiction.

Now we consider that $|V\left(H\right)|>0$ . Let $H={H}_{1}\cup {H}_{2}$ where H1 is the union of components of H which satisfies that ${d}_{G-S}\left(x\right)=k-1$ for any vertex $x\in V\left({H}_{1}\right)$ and ${H}_{2}=H-{H}_{1}$ . By Lemma 2.3, H1 has a maximal independent set I1 and the covering set ${C}_{1}=V\left({H}_{1}\right)-{I}_{1}$ such that

$|V\left({H}_{1}\right)|\le \left(k-\frac{1}{k+1}\right){i}^{\prime }+\sum _{j=0}^{k-2}\left(j+1\right){{i}^{″}}_{j},\text{\hspace{0.17em}}\text{\hspace{0.17em}}|{C}_{1}|\le \left(k-1-\frac{1}{k+1}\right){i}^{\prime }+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}j{{i}^{″}}_{j},$

where ${i}^{\prime }=|{I}^{\prime }|=|\left\{x|x\in {I}_{1},{d}_{{H}_{1}}\left(x\right)=k-1={d}_{G-S}\left(x\right)\right\}|$ , ${{i}^{″}}_{j}=|\left\{x|x\in {I}^{″}={I}_{1}-{{I}^{\prime }}_{1},{d}_{{H}_{1}}\left(x\right)=j<{d}_{G-S}\left(x\right)\right\}|$ .

On the other hand, we may assume that $\delta \left({H}_{2}\right)\ge 1$ . Since $\text{Δ}\left({H}_{2}\right)\le k-1$ , let ${T}_{j}=\left\{x\in {V\left({H}_{2}\right)|}_{G-S}\left(x\right)=j\right\}$ for $1\le j\le k-1$ . By the definition of H2 we know that there exists one vertex with degree at most $k-2$ in $G-S$ from each component of H2. According to Lemma 2.4, H2 has a maximal independent set I2 and the covering set ${C}_{2}=V\left({H}_{2}\right)-{I}_{2}$ such that

$\sum _{j=1}^{k-1}\left(k-j\right){c}_{j}\le \sum _{j=1}^{k-1}\left(k-2\right)\left(k-j\right){i}_{j}$ (2)

where ${c}_{j}=|{C}_{2}\cap {T}_{j}|$ and ${i}_{j}=|{I}_{2}\cap {T}_{j}|$ for $j=1,\cdots ,k-1$ .

Set $W=V\left(G\right)-S-T$ and $U=S\cup {C}_{1}\cup \left({N}_{G}\left({{I}^{″}}_{1}\right)\cap W\right)\cup {C}_{2}\cup \left({N}_{G}\left({I}_{2}\right)\cap W\right)$ . Then

$|U|\le |S|+|{C}_{1}|+\sum _{j=0}^{k-2}\left(k-1-j\right){{i}^{″}}_{j}+\sum _{j=0}^{k-1}\text{\hspace{0.17em}}j{i}_{j}.$

Let ${X}^{\prime }$ be the set of isolated vertices of $G-U$ , then . Set $X={X}^{\prime }\cup {C}_{1}\cup \cdots \cup {C}_{m}$ , then and ${N}_{G}\left(X\right)\subseteq U\cup {C}_{1}\cup \cdots \cup {C}_{m}$ , ${N}_{G}\left(X\right)\ne V\left(G\right)$ . By the definition of $bind\left(G\right)$ we have $|U|+mk\ge |{N}_{G}\left(X\right)|\ge bind\left(G\right)|X|$ . Therefore

(3)

From (1) we have $k\left({t}_{0}+m\right)+|V\left({H}_{1}\right)|+\sum _{j=1}^{k-1}\left(k-j\right){i}_{j}+\sum _{j=1}^{k-1}\left(k-j\right){c}_{j}\ge k|S|$ .

Combined with (3),

$\begin{array}{l}k\left({t}_{0}+m\right)+|V\left({H}_{1}\right)|+k|{C}_{1}|+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}k\left(k-1-j\right){{i}^{″}}_{j}+\sum _{j=1}^{k-1}\left(k-j\right){c}_{j}\\ >kbind\left(G\right)\left({t}_{0}+mk+{{i}^{\prime }}_{1}+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}{{i}^{″}}_{j}\right)+\sum _{j=1}^{k-1}\left(kbind\left(G\right)-kj-k+j\right){i}_{j}-m{k}^{2}.\end{array}$

By the notation of $bind\left(G\right)$ , we have

$\begin{array}{l}|V\left({H}_{1}\right)|+k|{C}_{1}|+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}k\left(k-1-j\right){{i}^{″}}_{j}+\sum _{j=1}^{k-1}\left(k-j\right){c}_{j}\\ >kbind\left(G\right)\left({{i}^{\prime }}_{1}+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}{{i}^{″}}_{j}\right)+\sum _{j=1}^{k-1}\left(kbind\left(G\right)-kj-k+j\right){i}_{j}.\end{array}$

By Lemma 2.3, we get that

$\begin{array}{l}|V\left({H}_{1}\right)|+k|{C}_{1}|+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}k\left(k-1-j\right){{i}^{″}}_{j}\\ \le \left(k-\frac{1}{k+1}+k\left(k-1-\frac{1}{k+1}\right)\right){{i}^{\prime }}_{1}+\sum _{j=0}^{k-2}\left(j+1+kj\right){{i}^{″}}_{j}+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}k\left(k-1-j\right){{i}^{″}}_{j}\\ =\left({k}^{2}-1\right){{i}^{\prime }}_{1}+\sum _{j=0}^{k-2}\left({k}^{2}-k+j+1\right){{i}^{″}}_{j}.\end{array}$

Combined with (2),

$\begin{array}{l}\sum _{j=1}^{k-1}\left(k-2\right)\left(k-j\right){i}_{j}+\left({k}^{2}-1\right){{i}^{\prime }}_{1}+\sum _{j=0}^{k-2}\left({k}^{2}-k+j+1\right){{i}^{″}}_{j}\\ >kbind\left(G\right)\left({{i}^{\prime }}_{1}+\sum _{j=0}^{k-2}\text{\hspace{0.17em}}{{i}^{″}}_{j}\right)+\sum _{j=1}^{k-1}\left(kbind\left(G\right)-kj-k+j\right){i}_{j}\\ \ge \left({k}^{2}-1\right){{i}^{\prime }}_{1}+kbind\left(G\right)\sum _{j=0}^{k-2}\text{\hspace{0.17em}}{{i}^{″}}_{j}+\sum _{j=1}^{k-1}\left(kbind\left(G\right)-kj-k+j\right){i}_{j}.\end{array}$

We have

Thus at least one of the following two cases must hold.

Case 1. There exists at least one j satisfying $\left(k-2\right)\left(k-j\right)>kbind\left(G\right)-kj-k+j$ . It follows that $bind\left(G\right)<\frac{{k}^{2}-k+j}{k}\le k-\frac{1}{k}$ ($j\le k-1$ ), a contradiction.

Case 2. ${k}^{2}-1>kbind\left(G\right)$ . In this case we have $bind\left(G\right) , a contradiction.

The proof of Theorem 1.3 is complete. $\square$

Remark. The result in Theorem 1.2 is sharp. To see this, consider the graph ${G}_{1}={K}_{n}\vee \left(n+1\right){K}_{1}$ where n is an arbitrary positive integer. We can immediately obtain that $bind\left({G}_{1}\right)=\frac{n}{n+1}<1$ and $bind\left({G}_{1}\right)$ is arbitrary close to 1 when n is enough. If we choose $S=V\left({K}_{n}\right)$ , then $i\left(G-S\right)=n+1>|S|$ . It follows that G1 has no fractional perfect matching by Lemma 2.2.

To see Theorem 1.3 is also sharp, we construct the following graph G2: If $k=2$ , let $V\left({G}_{2}\right)=V\left(A\right)\cup V\left(B\right)\cup V\left(C\right)$ where $A={K}_{\left(nk+1\right)\left(k-1\right)}={K}_{2n+1}$ , $B=\left(nk+1\right){K}_{1}=\left(2n+1\right){K}_{1}$ and $C={K}_{n\left(k-1\right)}={K}_{n}$ . Set other edges in G2 are a perfect matching between A and B and all the pairs between B and C. This follows that $bind\left({G}_{2}\right)=\frac{\left(nk+1\right)\left(k-1\right)+n\left(k-1\right)}{nk+1}\left(k=2\right)$ . It is easy to see that $bind\left({G}_{2}\right) and $bind\left({G}_{2}\right)$ can be made arbitrary close to $k-\frac{1}{k}$ when n is large enough ($k=2$ ). Let $S=C$ , by Lemma 2.1 G2 has no fractional k-factor ($k=2$ ).

4. Conclusion

Therefore, we conclude that a graph G has a fractional 1-factor if $bind\left(G\right)\ge 1$ and has a fractional k-factor if $bind\left(G\right)\ge k-\frac{1}{k}$ . Furthermore, it is showed that both results are best possible in some sense.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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