Binding Number and Fractional k-Factors of Graphs

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 $bind\left(G\right)=\min \left\{\frac{\left|N_G\left(X\right)\right|}{\left|X\right|}:\varnothing \ne X\subseteq V\left(G\right)\right\}$ . 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.

• Binding Number
• Fractional k-Factor
• Fractional Matching
• Independent Set
• Covering Set

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 ${\displaystyle {\sum }_{e\ni x}h\left(e\right)}=k$ 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

$bind\left(G\right)=\min \left\{\frac{\left|N_G\left(X\right)\right|}{\left|X\right|}:\varnothing \ne X\subseteq V\left(G\right)\right\}$. 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 $\left|N_G\left(X\right)\right|\ge c\left|X\right|$. 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 $\left|V\left(G\right)\right|\ge 2$, 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 $\left|V\left(G\right)\right|\ge k+1$ 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 $V\left(G\right)$,

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

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 $V\left(G\right)$,

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

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

$\left|V\left(H\right)\right|\le \left(k-\frac{1}{k+1}\right)i^{\prime }+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}\left(j+1\right){i^{″}}_j,\left|C\right|\le \left(k-1-\frac{1}{k+1}\right)i^{\prime }+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}j{i^{″}}_j,$

where $i^{\prime }=\left|I^{\prime }\right|=\left|\left\{x|x\in I,d_H\left(x\right)=k-1=d_G\left(x\right)\right\}\right|$, ${i^{″}}_j=\left|\left\{x|x\in I^{″}=I-I^{\prime },d_H\left(x\right)=j<d_G\left(x\right)\right\}\right|$.

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

$\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-j\right)c_j\le \underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-2\right)\left(k-j\right)i_j$

where $c_j=\left|C\cap T_j\right|$ and $i_j=\left|I\cap T_j\right|$ 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 $i\left(G-S\right)>\left|S\right|$. We choose X as the set of isolated vertices of $G-S$, that is, $\left|X\right|=i\left(G-S\right)$. Since G is connected, it follows that $S\ne \varnothing$, and $\left|N_G\left(X\right)\right|\subseteq S\ne V\left(G\right)$.

According to the definition of $bind\left(G\right)$, we obtain $bind\left(G\right)\le \frac{\left|N_G\left(X\right)\right|}{\left|X\right|}\le \frac{\left|S\right|}{i\left(G-S\right)}<1$, 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

$k\left|T\right|-d_{G-S}\left(T\right)>k\left|S\right|,$(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 $\left|V\left(H\right)\right|=0$, by (1) we get $k\left|T_0\right|+mk>k\left|S\right|$, that is, $\left|S\right|\le \left|T_0\right|+m-1$.

If $m=0$, $\left|S\right|<\left|T_0\right|$. Set $X=T_0$, then $\left|X\right|=\left|T_0\right|$ and $N_G\left(X\right)\subseteq S$, $N_G\left(X\right)\ne V\left(G\right)$. We obtain that $bind\left(G\right)\le \frac{\left|N_G\left(X\right)\right|}{\left|X\right|}\le \frac{\left|S\right|}{\left|T_0\right|}<1$, 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 $\left|X\right|=\left(m-1\right)k+1+\left|T_0\right|$ 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

$bind\left(G\right)\le \frac{\left|N_G\left(X\right)\right|}{\left|X\right|}\le \frac{mk-1+\left|S\right|}{\left(m-1\right)k+1+\left|T_0\right|}\le \frac{mk-1+m-1+\left|T_0\right|}{\left(m-1\right)k+1+\left|T_0\right|}.$

When $k\ge 2$,

$\begin{array}{l}\left(\left(m-1\right)k+1+\left|T_0\right|\right)\left(k-\frac{1}{k}\right)-\left(mk-1+m-1+\left|T_0\right|\right)\\ \ge \left(\left(m-1\right)k+1\right)\left(k-\frac{1}{k}\right)-\left(mk-1+m-1\right)=\left(m-1\right)\left(k^2-k-2\right)+1-\frac{1}{k}>0.\end{array}$

That is, $bind\left(G\right)<k-\frac{1}{k}$, a contradiction.

Now we consider that $\left|V\left(H\right)\right|>0$. 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

$\left|V\left(H_1\right)\right|\le \left(k-\frac{1}{k+1}\right)i^{\prime }+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}\left(j+1\right){i^{″}}_j,\left|C_1\right|\le \left(k-1-\frac{1}{k+1}\right)i^{\prime }+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}j{i^{″}}_j,$

where $i^{\prime }=\left|I^{\prime }\right|=\left|\left\{x|x\in I_1,d_{H_1}\left(x\right)=k-1=d_{G-S}\left(x\right)\right\}\right|$, ${i^{″}}_j=\left|\left\{x|x\in I^{″}=I_1-{I^{\prime }}_1,d_{H_1}\left(x\right)=j<d_{G-S}\left(x\right)\right\}\right|$.

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

$\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-j\right)c_j\le \underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-2\right)\left(k-j\right)i_j$(2)

where $c_j=\left|C_2\cap T_j\right|$ and $i_j=\left|I_2\cap T_j\right|$ 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

$\left|U\right|\le \left|S\right|+\left|C_1\right|+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}\left(k-1-j\right){i^{″}}_j+\underset{j=0}{\overset{k-1}{{\displaystyle \sum }}}j{i}_j.$

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 $\left|U\right|+mk\ge \left|N_G\left(X\right)\right|\ge bind\left(G\right)\left|X\right|$. Therefore

(3)

From (1) we have $k\left(t_0+m\right)+\left|V\left(H_1\right)\right|+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-j\right)i_j+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-j\right)c_j\ge k\left|S\right|$.

Combined with (3),

$\begin{array}{l}k\left(t_0+m\right)+\left|V\left(H_1\right)\right|+k\left|C_1\right|+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}k\left(k-1-j\right){i^{″}}_j+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-j\right)c_j\\ >kbind\left(G\right)\left(t_0+mk+{i^{\prime }}_1+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}{i^{″}}_j\right)+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(kbind\left(G\right)-kj-k+j\right)i_j-m{k}^2.\end{array}$

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

$\begin{array}{l}\left|V\left(H_1\right)\right|+k\left|C_1\right|+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}k\left(k-1-j\right){i^{″}}_j+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-j\right)c_j\\ >kbind\left(G\right)\left({i^{\prime }}_1+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}{i^{″}}_j\right)+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(kbind\left(G\right)-kj-k+j\right)i_j.\end{array}$

By Lemma 2.3, we get that

$\begin{array}{l}\left|V\left(H_1\right)\right|+k\left|C_1\right|+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}k\left(k-1-j\right){i^{″}}_j\\ \le \left(k-\frac{1}{k+1}+k\left(k-1-\frac{1}{k+1}\right)\right){i^{\prime }}_1+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}\left(j+1+kj\right){i^{″}}_j+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}k\left(k-1-j\right){i^{″}}_j\\ =\left(k^2-1\right){i^{\prime }}_1+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}\left(k^2-k+j+1\right){i^{″}}_j.\end{array}$

Combined with (2),

$\begin{array}{l}\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(k-2\right)\left(k-j\right)i_j+\left(k^2-1\right){i^{\prime }}_1+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}\left(k^2-k+j+1\right){i^{″}}_j\\ >kbind\left(G\right)\left({i^{\prime }}_1+\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}{i^{″}}_j\right)+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(kbind\left(G\right)-kj-k+j\right)i_j\\ \ge \left(k^2-1\right){i^{\prime }}_1+kbind\left(G\right)\underset{j=0}{\overset{k-2}{{\displaystyle \sum }}}{i^{″}}_j+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}\left(kbind\left(G\right)-kj-k+j\right)i_j.\end{array}$

We have

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.

References

1. Bondy, J. and Murty, U. (1976) Graph Theory with Applicatins. American Elsevier.

2. Scheinermman, E. and Ullman, D. (1997) Fractional Graph Theory. John and Wiley and Sons, Inc.

3. Woodall, D.R. (1973) The Binding Number of a Graph and Its Anderson Number. Journal of Combinatorial Theory, Series B, 15, 225-255. >https://doi.org/10.1016/0095-8956(73)90038-5

4. Huilgol, M.I. and Divya, B. (2022) Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs. Open Journal of Discrete Mathematics, 12, 1-16. >https://doi.org/10.4236/ojdm.2022.121001

5. Burchett, P.A. (2020) The Number of Minimum Roman and Minimum Total Dominating Sets for Some Chessboard Graphs. Open Journal of Discrete Mathematics, 10, 31-44. >https://doi.org/10.4236/ojdm.2020.101004

6. Hong, X., Ao, G. and Gao, F. (2021) The Generalization of Signed Domination Number of Two Classes of Graphs. Open Journal of Discrete Mathematics, 11, 114-132. >https://doi.org/10.4236/ojdm.2021.114009

7. Anderson, I. (1971) Perfect Matchings of a Graph. Journal of Combinatorial Theory, Series B, 10, 183-186. >https://doi.org/10.1016/0095-8956(71)90041-4

8. Plummer, M.D. and Saito, A. (2017) Toughness, Binding Number and Restricted Matching Extension in a Graph. Discrete Mathematics, 340, 2665-2672. >https://doi.org/10.1016/j.disc.2016.10.003

9. Katerinis, P. and Woodall, D.R. (1987) Binding Numbers of Graphs and the Existence of k-Factors. The Quarterly Journal of Mathematics, 38, 221-228. >https://doi.org/10.1093/qmath/38.2.221

10. Katerinis, P. (1985) Minimum Degree of a Graph and the Existence of k-Factors. Proceedings of the Indian Academy of Sciences-Section A, 94, 123-127. >https://doi.org/10.1007/bf02880991

11. Zhou, S.Z. (2013) Binding Numbers for Fractional ID-k-Factor-Critical Graphs. Acta Mathematica Sinica, English Series, 30, 181-186. >https://doi.org/10.1007/s10114-013-1396-9

12. Kano, M. and Tokushige, N. (1992) Binding Numbers and F-Factors of Graphs. Journal of Combinatorial Theory, Series B, 54, 213-221. >https://doi.org/10.1016/0095-8956(92)90053-z

13. Yashima, T. (2018) Binding Number, Minimum Degree and (g, f)-Factors of Graphs, Contrib. Discrete Mathematics, 13, 137-141.

14. Zhou, S. (2021) Binding Numbers and Restricted Fractional (g, f)-Factors in Graphs. Discrete Applied Mathematics, 305, 350-356. >https://doi.org/10.1016/j.dam.2020.10.017

15. Wu, J. and Gao, W. (2018) Binding Number Condition for Fractional (g, f, n’, m)-Critical Deleted Graph in the New Setting. Utilitas Mathematica, 109, 129-137.

16. Chen, Y. and Dai, G. (2022) Binding Number and Path-Factor Critical Deleted Graphs. AKCE International Journal of Graphs and Combinatorics, 19, 197-200. >https://doi.org/10.1080/09728600.2022.2094299

17. Fan, D. and Lin, H. (2024) Binding Number, k-Factor and Spectral Radius of Graphs. The Electronic Journal of Combinatorics, 31, Article No. 1.30. >https://doi.org/10.37236/12165

18. Feng, X. and Deng, X. (2023) Toughness and Binding Number Bounds of Star-Like and Path Factor. RAIRO-Operations Research, 57, 1167-1177. >https://doi.org/10.1051/ro/2023057

19. Liu, G. and Zhang, L. (2001) Fractional (g, f)-Factors of Graphs. Acta Mathematica Scientia, 21, 541-545. >https://doi.org/10.1016/s0252-9602(17)30443-5

20. Chang, R. and Zhu, Y. (2012) Toughness and [a, b]-Factors in Graphs. Ars Combinatoria, 105, 451-459.

21. Liu, G. and Zhang, L. (2008) Toughness and the Existence of Fractional k-Factors of Graphs. Discrete Mathematics, 308, 1741-1748. >https://doi.org/10.1016/j.disc.2006.09.048