Factors and connected factors in tough graphs with high isolated toughness

TL;DR

This paper establishes conditions under which graphs with high isolated toughness contain specific factors and connected factors, extending understanding of graph structure related to toughness and factorization.

Contribution

It introduces new sufficient conditions for the existence of particular factors and connected factors in graphs with high isolated toughness.

Findings

Graphs satisfying certain isolated toughness conditions have specific factors.

Conditions for the existence of factors with prescribed degrees are established.

Connected factors with degrees in a specified set exist under given toughness constraints.

Abstract

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. Assume that for all $S\subseteq V(G)$, $$\sum_{v\in I(G\setminus S)}f(v)(f(v)+1)\le |S|,$$ where $I(G\setminus S)$ denotes the set of isolated vertices of $G\setminus S$. In this paper, we show that if for all $S\subseteq V(G)$, $$\omega(G\setminus S)\le \sum_{v\in S}(f(v)-1)+1,$$ and $\sum_{v\in V(G)}f(v)$ is even, then $G$ has a factor $F$ such that for each vertex $v$, $d_F(v)=f(v)$, where $\omega(G\setminus S)$ denotes the number of components of $G\setminus S$. Moreover, we show that if for all $S\subseteq V(G)$, $$\omega(G\setminus S)\le \frac{1}{4}|S|+1,$$ and $f\ge 2$, then $G$ has a connected factor $H$ such that for each vertex $v$, $d_H(v)\in \{f(v),f(v)+1\}$.

Equations

$$\sum _{v\in I(G\setminus S)}f(v)(f(v)+1)\le |S|,$$

$$\omega (G\setminus S)\le \sum _{v\in S}(f(v)-1)+1,$$

$$\omega (G\setminus S)\le \frac{1}{4}|S|+1,$$

$${\omega }_f(G,A,B)\le \sum _{v\in A}f(v)+\sum _{v\in B}(d_{G\setminus A}(v)-f(v))+1.$$

$$\omega (G\setminus (A\cup B))\le \sum _{v\in A}f(v)+\sum _{v\in B}(d_{G\setminus A}(v)-f(v))+1,$$

$$\omega (G\setminus (A\cup B))\le \omega (G\setminus (A\cup B_u))+d-1\le q(A,B_u)+d-1=q(A,B)-d_{G[B]}(u)+g(u)-1\le q(A,B).$$

$$\omega (G\setminus (A\cup B))=\omega (G\setminus (A_u\cup B_u))\le q(A_u,B_u)=q(A,B)-d_{G\setminus A}(u)+f(u)+g(u)\le q(A,B),$$

$$\sum _{v\in V(H)}\phi (v)\le \sum _{v\in I}\phi (v)(d_H(v)+1).$$

$$\sum _{v\in V(H)\setminus I}\phi (v)=\sum _{u\in I}\sum _{v\in N(u)}\phi (v)\le \sum _{u\in I}\phi (u)d_H(u).$$

$$\sum _{v\in V(H)}(\phi (v)-d(v))\le \sum _{v\in I}(d(v)+1)(\phi (v)-d(v)).$$

$$0\le \sum _{v\in A}f(v)+\sum _{v\in B}(d_{G\setminus A}(v)-g(v)).$$

$$t(v)=\{\begin{array}{ll}g(v)(1+\frac{1}{a})-1,& \text{when g(v)≤a+2};\\ \frac{1}{4a}((g+a+1{)}^2-{\text{ε}}_0(v))-1,& \text{otherwise},\end{array}$$

$$\sum _{v\in B}(g(v)-d(v))\le \sum _{v\in I}(d(v)+1)(g(v)-d(v)).$$

$$(d(v)+1)\big{(}g(v)-d(v)\big{)}+a\,d(v)=(d(v)+1)\big{(}g(v)+a-d(v)\big{)}-a\leq a\,t(v),$$

$$\sum_{v\in B}(g(v)-d(v))\leq\sum_{v\in I}(d(v)+1)\big{(}g(v)-d(v)\big{)}\leq\sum_{v\in I}\big{(}a\,t(v)-a\,d(v)\big{)}\leq a|A|.$$

$$0\le a|A|+\sum _{v\in B}(d(v)-g(v))\le \sum _{v\in A}f(v)+\sum _{v\in B}(d(v)-g(v)).$$

$$\omega (G\setminus S)<\frac{1}{n}|S|+2,$$

$$(k+\frac{1}{n})|B_i|\le (k+\frac{1}{n})iso(G\setminus S_i)\le |S_i|\le |A|+\sum _{v\in B_i}{d}_{G\setminus A}(v),$$

$$(k+\frac{1}{n})|B|=\sum _{1\le i<k}(k+\frac{1}{n})|B_i|\le (k-1)|A|+\sum _{1\le i<k}\sum _{v\in B_i}{d}_{G\setminus A}(v)=(k-1)|A|+\sum _{v\in B}{d}_{G\setminus A}(v).$$

$$\omega (G\setminus A\cup B)<\frac{1}{n}(|A|+|B|)+2.$$

$$\omega (G\setminus (A\cup B))<(k-1+\frac{1}{n})|A|+\sum _{v\in B}(d_{G\setminus A}(v)-k)+2\le k|A|+\sum _{v\in B}(d_{G\setminus A}(v)-k)+2.$$

$$\omega (G\setminus S)<\sum _{v\in S}(f(v)-\text{ε})+2,$$

$$\sum _{v\in B}(\phi (v)-d(v))\le \sum _{v\in I}(d(v)+1)(\phi (v)-d(v)).$$

$$(d(v)+1)\big{(}\varphi(v)-d(v)\big{)}+\text{$\varepsilon$}\,d(v)=(d(v)+1)\big{(}2f(v)-d(v)\big{)}-\text{$\varepsilon$}\leq\text{$\varepsilon$}\,t(v),$$

$$\sum_{v\in B}(\varphi(v)-d(v))\leq\sum_{v\in I}(d(v)+1)\big{(}\varphi(v)-d(v)\big{)}\leq\sum_{v\in I}\big{(}\text{$\varepsilon$}\,t(v)-\text{$\varepsilon$}d(v)\big{)}\leq\text{$\varepsilon$}|A|.$$

$$\omega (G\setminus (A\cup B))<\sum _{v\in A\cup B}(f(v)-\text{ε})+2=\sum _{v\in A}(f(v)-\text{ε})+\sum _{v\in B}(f(v)-\text{ε})+2.$$

$$\omega (G\setminus (A\cup B))\le \sum _{v\in A}f(v)+\sum _{v\in B}(d(v)-f(v))+1.$$

$$\omega (G\setminus S)\le \sum _{v\in S}(f(v)-1)+1,$$

$$\omega (G\setminus S)<\sum _{v\in S}(f(v)-a)+2,$$

$$\sum _{v\in B}(\phi (v)-d(v))\le \sum _{v\in I}(d(v)+1)(\phi (v)-d(v)).$$

Full text

[TABLE]

where $I(G\setminus S)$ denotes the set of isolated vertices of $G\setminus S$. In this paper, we show that if for all $S\subseteq V(G)$,

[TABLE]

and $\sum_{v\in V(G)}f(v)$ is even, then $G$ has a factor $F$ such that for each vertex $v$, $d_{F}(v)=f(v)$, where $\omega(G\setminus S)$ denotes the number of components of $G\setminus S$. Moreover, we show that if for all $S\subseteq V(G)$,

[TABLE]

and $f\geq 2$, then $G$ has a connected factor $H$ such that for each vertex $v$, $d_{H}(v)\in\{f(v),f(v)+1\}$.

*Keywords:

Toughness; isolated toughness; regular factor; connected factor; $f$-factor. *

1 Introduction

In this article, all graphs have no loop, but multiple edges are allowed and a simple graph is a graph without multiple edges. Let $G$ be a graph. The vertex set and the edge set of $G$ are denoted by $V(G)$ and $E(G)$, respectively. We also denote by $iso(G)$, $odd(G)$, and $\omega(G)$ the number of isolated vertices of $G$, the number of components of $G$ with odd number of vertices, and the number of components of $G$, respectively. For a vertex set $S$ of $G$, we denote by $G[S]$ the induced subgraph of $G$ with the vertex set $S$ containing precisely those edges of $G$ whose ends lie in $S$. The vertex set $S$ is called an independent set, if there is no edge of $G$ connecting vertices in $S$. The maximum size of all independent sets of $G$ is denoted by $\alpha(G)$. 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 $S\subseteq V(G)$. Furthermore, $G$ is said to be $t$-iso-tough, if $iso(G\setminus S)\leq\frac{1}{t}|S|$ for all $S\subseteq V(G)$. This definition is a little different from [13, 14] for the sake of simplicity. Note that when $G$ is $t$-iso-tough, for each vertex $v$, the number of its neighbours must be at least $t$ and hence the conditions $d_{G}(v)\geq t$ and $V(G)\geq t+1$ must automatically hold. More generally, when $t$ is a real function on $V(G)$, we say that $G$ is $t$-iso-tough, if for all $S\subseteq V(G)$, $\sum_{v\in I(G\setminus S)}t(v)\leq|S|$, where $I(G\setminus S)$ denotes the set of all isolated vertices of $G\setminus S$. We denote by $N_{G}(I)$ the set of all neighbours of vertices of $I$ in $G$. For a set $A$ of integers, an $A$-factor is a spanning subgraph with vertex degrees in $A$. Let $g$ and $f$ be two integer-valued functions on $V(G)$. A $(g,f)$-factor of $G$ is a spanning subgraph $F$ such that for each vertex $v$, $g(v)\leq d_{F}(v)\leq f(v)$. When $g=f-1$, we call it a $\{g,g+1\}$-factor as well. An $f$-factor of $G$ refers to a spanning subgraph $F$ such that for each vertex $v$, $d_{F}(v)=f(v)$. A near $f$-factor refers to a spanning subgraph $F$ such that for each vertex $v$, $d_{F}(v)=f(v)$, except for at most one vertex $u$ with $d_{F}(u)=f(u)+1$. Note that when the sum of all $f(v)$ taken over all vertices $v$ is even, $F$ is a near $f$-factor if and only if $F$ is an $f$-factor. Note that several theorems in graph theory for the existence of $f$-factors can be developed to a near $f$-factor version. This type of factors is useful when a factor is required for extending to connected factors with bounded degrees as the proof of Theorem 4.8. For example, see [3, 9]. For convenience, we write $\min f$ for $\min\{f(v):v\in V(G)\}$ and write $\max f$ for $\max\{f(v):v\in V(G)\}$. Let $A$ and $B$ be two disjoint vertex sets. We denote by $\omega_{f}(G,A,B)$ the number of components $C$ of $G\setminus(A\cup B)$ satisfying $\sum_{v\in V(C)}f(v)\not\stackrel{{\scriptstyle 2}}{{\equiv}}d_{G}(C,B)$, where $d_{G}(C,B)$ denotes the number of edges of $G$ with one end in $V(C)$ and the other one in $B$. Throughout this article, all variables $k$ are positive integers.

In 1947 Tutte introduced the following criterion for the existence of a perfect matching.

Theorem 1.1

.([16])* A graph $G$ has a $1$-factor if and only if for all $S\subseteq V(G)$, $odd(G\setminus S)\leq|S|$.*

In 1978 Vergenas formulated a criterion for the existence of $(1,f)$-factors and showed that the criterion becomes simpler for the following special case.

Theorem 1.2

.([11])* Let $G$ be a graph and let $f$ be an integer-valued function on $V(G)$ with $f\geq 2$. Then $G$ has a $(1,f)$-factor if and only if for all $S\subseteq V(G)$, $iso(G\setminus S)\leq\sum_{v\in S}f(v).$*

In 1985 Enomoto, Jackson, Katerinis, and Saito proved the following theorem on tough graphs, which was originally conjectured by Chvátal (1973) [5]. In 1990 Katerinis [10] generalized their result by replacing a weaker sufficient toughness condition for the existence of $[a,b]$-factors, provided that $a>b$.

Theorem 1.3

.([8])* Every $k$-tough graph $G$ of order at least $k+1$ with $k|V(G)|$ even has a $k$-factor.*

In 2007 Ma and Yu strengthened Katerinis’ result by replacing isolated toughness condition as the following theorem. In this paper, we provide a supplement for their result by improving Theorem 1.3 for $(k+1)$-iso-tough graphs by showing that the needed toughness can be pushed down to the fixed number $1$. In Section 5, we also establish another refined version in $k$-iso-tough graphs.

Theorem 1.4

.([13])* Every $(a-\frac{b-a}{b})$-iso-tough graph has an $[a,b]$-factor, when $b>a\geq 1$.*

In 1973 Chvátal [5] 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 2000 Ellingham and Zha [7] confirmed a weaker version of this conjecture by proving that every $4$-tough graph of order at least three admits a connected $\{2,3\}$-factor. Motivated by this result, one way ask whether higher toughness can guarantee the existence of connected $\{k,k+1\}$-factors. The following theorem shows that the answer is positive. In this paper, we show that the needed toughness of this theorem can be pushed down to the fixed number $3$ but in $(k+1)$-iso-tough graphs.

Theorem 1.5

.([6, 7], see [9])* Every $k$-tough graph of order at least $k+1$ has a connected $\{k,k+1\}$-factor, where $k\geq 3$.*

In 1990 Katerinis formulated the following sufficient toughness condition for the existence of $f$-factors. In Section 4, we introduce some sufficient toughness conditions for the existence of $f$-factors and connected $\{f,f+1\}$-factors in graphs with high enough isolated toughness as mentioned in the abstract.

Theorem 1.6

.([10])* Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$ satisfying $a\leq f\leq b$, where $a$ and $b$ are two positive integers. If $G$ is $\frac{1}{4a}((b+a)^{2}+2(b-a)+1)$-tough and $\sum_{v\in V(G)}f(v)$ is even, then $G$ has an $f$-factor.*

2 Tools and preliminary results

In this section, we shall provide some necessary tools for applying in the next sections. Before doing so, let us recall a theorem due to Tutte (1952) as the following version.

Theorem 2.1

.([17])* Let $G$ be a general graph and let $f$ be an integer-valued function on $V(G)$. Then $G$ has a near $f$-factor if and only if for all disjoint subsets $A$ and $B$ of $V(G)$,*

[TABLE]

The following corollary is an application of Theorem 2.1, which is inspired by Lemma 4 in [10].

Corollary 2.2

.(see [10])* Let $G$ be a general graph and let $f$ be an integer-valued function on $V(G)$. Then $G$ has a near $f$-factor, if*

[TABLE]

for all disjoint subsets $A$ and $B$ of $V(G)$ satisfying $d_{G[B]}(u)\leq f(u)-2$ and $d_{G\setminus A}(u)\leq 2f(u)-1$ for each $u\in B$.

**Proof. **

Let us define $g=f$. We are going to show that the inequality holds for any two disjoint subsets $A$ and $B$ of $V(G)$ and so the proof follows from Theorem 2.1. By induction on $|B|$. Let $q(A,B)$ be the right-hand side of the inequality in the corollary. Assume that $B$ has a vertex $u$ with $d_{G[B]}(u)\geq g(u)-1$ or $d_{G\setminus A}(u)\geq f(u)+g(u)$. Define $B_{u}=B\setminus\{u\}$. If $d_{G[B]}(u)\geq g(u)-1$, then

[TABLE]

where $d$ denotes the number of edges of $G$ incident to $u$ with the other end in $V(G)\setminus(A\cup B)$. Also, if $d_{G\setminus A}(u)\geq f(u)+g(u)$, then

[TABLE]

where $A_{u}=A\cup\{u\}$. Hence the lemma holds.

• $\Box$*

The following theorem can generalize Lemma 1 in [10] and plays an essential role in this paper.

Theorem 2.3

.* Let $H$ be a graph. If $\varphi$ is a nonnegative real function on $V(H)$, then there is a maximal independent subset $I$ of $V(H)$ such that*

[TABLE]

**Proof. **

Define $H_{0}=H$. For every nonnegative integer $i$ with $|V(H_{i})|\geq 1$, take $u_{i}$ to be a vertex of $H_{i}$ with the maximum $\varphi(u_{i})$ and set $H_{i+1}=H_{i}\setminus(N(u_{i})\cup\{u_{i}\})$, where $N(u_{i})$ denotes the set of all neighbours of $u_{i}$ in $H_{i}$. Define $I$ to be the set of all selected vertices $u_{i}$. It is not hard to check that $I$ is a maximal independent set of $H$ and $\{N(u)\cup\{u\}:u\in I\}$ is a partition of $V(H)$. Since $\varphi(u)\geq 0$,

[TABLE]

This inequality can complete the proof.

• $\Box$*

Corollary 2.4

.([4, 15])* For every graph $H$, we have $\alpha(H)\geq\sum_{v\in V(H)}\frac{1}{1+d_{H}(v)}.$*

**Proof. **

Apply Theorem 2.3 with $\varphi(v)=1/(1+d_{H}(v))$.

• $\Box$*

The following corollary provides an equivalent version for Theorem 2.3.

Corollary 2.5

.* Let $H$ be a graph and let $\varphi$ and $d$ be two real functions on $V(H)$. If for each $v\in V(H)$, $\varphi(v)\geq d(v)\geq d_{H}(v)$, then there is a maximal independent subset $I$ of $V(H)$ such that*

[TABLE]

**Proof. **

Apply Theorem 2.3 with replacing $(\varphi(v)-d(v))$ instead of $\varphi(v)$.

• $\Box$*

3 Isolated toughness and the existence of $\{f,f+1\}$-factors

Our aim in this section is to generalize Theorem 1.4 by giving isolated toughness conditions for existence of $(g,f)$-factors, provided that $g<f$. For this purpose, we need the following lemma due to Lovász (1970).

Lemma 3.1

.([12])* Let $G$ be a graph and let $g$ and $f$ be two integer-valued functions on $V(G)$ with $g<f$. Then $G$ has a $(g,f)$-factor, if and only if for all disjoint subsets $A$ and $B$ of $V(G)$,*

[TABLE]

The following theorem provides a common generalization for both of Theorems 2 and 3 in [13].

Theorem 3.2

.* Let $G$ be a graph and let $g$ and $f$ be two nonnegative integer-valued functions on $V(G)$ with $g<f$. Let $a$ be positive real number with $a\leq f$. Then $G$ has a $(g,f)$-factor, if it $t$-iso-tough, where for each vertex $v$,*

[TABLE]

in which \text{\varepsilon}_{0}(v)\in\{0,1\} such that \text{\varepsilon}_{0}(v)=1 if and only if $g(v)$ and $a$ are integers with the same parity.

**Proof. **

Let $A$ and $B$ be two disjoint subsets of $V(G)$. In order to apply Lemma 3.1, we should prove the inequality $0\leq\sum_{v\in A}f(v)+\sum_{v\in B}(d(v)-g(v))$, where $d(v)=d_{G\setminus A}(v)$. For this purpose, we may assume that for each $v\in B$, $d(v)\leq g(v)-1$. By applying Corollary 2.5 with $\varphi=g$, the graph $G[B]$ has an independent set $I$ such that

[TABLE]

Since $G$ is $t$-iso-tough, we have $\sum_{v\in I}t(v)\leq|A\cup N_{G}(I)|\leq|A|+\sum_{v\in I}d(v)$. Since $d(v)$ is integer, we must have

[TABLE]

regardless of $g(v)-1\leq(g(v)+a)/2$ or not. This implies that

[TABLE]

Therefore,

[TABLE]

Hence the assertion follows from Lemma 3.1.

• $\Box$*

When we consider the special case $\max g<\min f$, the theorem becomes simpler as the following result.

Corollary 3.3

.* Let $G$ be a graph and let $g$ and $f$ be two nonnegative integer-valued functions on $V(G)$ with $\max g<\min f$. If $G$ is $(g-1+\frac{g}{\min f})$-iso-tough, then it has a $(g,f)$-factor.*

**Proof. **

Apply Theorem 3.2 with $a=\min f$.

• $\Box$*

Corollary 3.4

.* Every $f(f+1)$-iso-tough graph $G$ has an $\{f,f+1\}$-factor, where $f$ is a nonnegative integer-valued function on $V(G)$.*

**Proof. **

Apply Theorem 3.2 with setting and $g^{\prime}=f$, $f^{\prime}=f+1$, and $a=1$.

• $\Box$*

4 Toughness, isolated toughness, and the existence of $f$-factors

In this section, we are going to present some sufficient toughness conditions for the existence of $f$-factors in graphs with high enough isolated toughness.

4.1 Regular factors in $1$-tough graphs

The following theorem significantly improves the needed toughness in Theorem 1.3 in graphs with a bit higher isolated toughness.

Theorem 4.1

.* Let $k$ be a positive integer and let $n$ be a real number with $n\geq 1$. If $G$ is a $(k+1/n)$-iso-tough graph and for all $S\subseteq V(G)$,*

[TABLE]

then $G$ has a near $k$-factor.

**Proof. **

Let $A$ and $B$ be two disjoint subsets of $V(G)$ such that for each $v\in B$, $d_{G[B]}(v)\leq k-2$. By applying a greedy coloring, one can decompose $B$ into $k-1$ independent vertex sets $B_{1},\ldots,B_{k-1}$. Let $S_{i}=A\cup N_{G}(B_{i})$. By the assumption, we must have

[TABLE]

which implies that

[TABLE]

By the assumption, we must also have

[TABLE]

Therefore,

[TABLE]

Thus the assertion follows from Corollary 2.2.

• $\Box$*

Remark 4.2

. Note that when $G$ has no complete subgraphs of order $k-1$ and $k\geq 5$, independent sets $B_{i}$ could be chosen such that $B_{k-1}=\emptyset$ using Brooks’ Theorem [2]. This fact allows us to reduce the lower bound on $n$ to $1/2$.

4.2 Graphs with toughness less than $1$

The following theorem gives a sufficient toughness condition for the existence of $f$-factors.

Theorem 4.3

.* Let $\varepsilon$ be a real number with 0<\text{\varepsilon}\leq 1. Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. If $G$ is f(f+1)/\text{\varepsilon}-iso-tough and for all $S\subseteq V(G)$,*

[TABLE]

then $G$ has a near $f$-factor.

**Proof. **

Let $A$ and $B$ be two disjoint subsets of $V(G)$. We may assume that $d(v)\leq 2f(v)-1$ for each $v\in B$, where $d(v)=d_{G\setminus A}(v)$. For each $v\in B$, define \varphi(v)=2f(v)-\text{\varepsilon} so that $\varphi(v)\geq d(v)$. By Corollary 2.5, the graph $G[B]$ has an independent set $I$ such that

[TABLE]

For each vertex $v$, define t(v)=f(v)(f(v)+1)/\text{\varepsilon}-1. Since $G$ is $t$-iso-tough, we have $\sum_{v\in I}t(v)\leq|A\cup N_{G}(I)|\leq|A|+\sum_{v\in I}d(v)$. Since $d(v)$ is integer, we must have

[TABLE]

which implies that

[TABLE]

On the other hand, by the assumption,

[TABLE]

Therefore,

[TABLE]

Hence the assertion follows from Corollary 2.2.

• $\Box$*

Corollary 4.4

.* Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. If $G$ is $f(f+1)$-iso-tough and for all $S\subseteq V(G)$,*

[TABLE]

then $G$ has a near $f$-factor.

**Proof. **

Apply Theorem 4.3 with \text{\varepsilon}=1.

• $\Box$*

The isolated toughness needed in Corollary 4.4 can be improved by a coefficient for graphs with higher toughness as the next theorem, provided that $\min f$ is sufficiently large.

Theorem 4.5

.* Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$, and let $a$ be a positive real number with $f\geq a\geq 1$. If $G$ is $\frac{1}{a}(f+a/2)^{2}$-iso-tough and for all $S\subseteq V(G)$,*

[TABLE]

then $G$ has a near $f$-factor.

**Proof. **

Let $A$ and $B$ be two disjoint subsets of $V(G)$. We may assume that $d(v)\leq 2f(v)-1$ for each $v\in B$, where $d(v)=d_{G\setminus A}(v)$. For each $v\in B$, define $\varphi(v)=2f(v)-1$ so that $\varphi(v)\geq d(v)$. By Corollary 2.5, the graph $G[B]$ has an independent set $I$ such that

[TABLE]

For each vertex $v$, define $t(v)=\frac{1}{a}((f(v)+a/2)^{2})-1$. Since $G$ is $t$-iso-tough, we have $\sum_{v\in I}t(v)\leq|A\cup N_{G}(I)|\leq|A|+\sum_{v\in I}d(v)$. In addition, we must have

[TABLE]

which implies that

[TABLE]

On the other hand, by the assumption,

[TABLE]

Therefore,

[TABLE]

Hence the assertion follows from Corollary 2.2.

• $\Box$*

4.3 Applications to the existence of connected $\{f,f+1\}$-factors

The following lemma is a useful tool for extending factors to connected factors by inserting a matching.

Lemma 4.6

.([7], see [9])* Let $\varepsilon$ be a real number with 0<\text{\varepsilon}\leq 2. Let $G$ be a simple graph and let $F$ be a factor of $G$ with minimum degree at least 2/\text{\varepsilon}+1. If for all $S\subseteq V(G)$,*

[TABLE]

then $G$ has a connected factor $H$ containing $F$ such that for each vertex $v$, $d_{H}(v)\in\{d_{F}(v),d_{F}(v)+1\}$, and $d_{H}(u)=d_{F}(u)$ for an arbitrary given vertex $u$.

The following result shows an application of Lemma 4.6 and Theorem 4.1.

Theorem 4.7

.* Every $3$-tough $(k+1/3)$-iso-tough graph has a connected $\{k,k+1\}$-factor, where $k\geq 3$.*

**Proof. **

We may assume that $G$ simple, by deleting multiple edges from $G$ (if necessary). By Theorem 4.1, the graph $G$ has a near $k$-factor $F$ so that for all vertices $v$, $d_{F}(v)=k$, except for at most one vertex $u$ with $d_{F}(u)=k+1$. By applying Lemma 4.6 with \text{\varepsilon}=1, the graph $G$ has a connected factor $H$ such that for each vertex $v$, $d_{H}(v)\in\{d_{F}(v),d_{F}(v)+1\}$, and also $d_{H}(u)=d_{F}(u)$. This implies that $H$ is a connected $\{k,k+1\}$-factor.

• $\Box$*

The next result shows an application of Lemma 4.6 and Corollary 4.4.

Theorem 4.8

.* Let $\varepsilon$ be a real number with 0<\text{\varepsilon}\leq 2. Let $G$ be graph and let $f$ be a positive integer-valued function on $V(G)$ with f\geq 2/\text{\varepsilon}+1. If $G$ is $f(f+1)$-iso-tough and for all $S\subseteq V(G)$,*

[TABLE]

then $G$ has a connected $\{f,f+1\}$-factor.

**Proof. **

We may assume that $G$ simple, by deleting multiple edges from $G$ (if necessary). By Corollary 4.4, the graph $G$ has a near $f$-factor $F$ so that for all vertices $v$, $d_{F}(v)=f(v)$, except for at most one vertex $u$ with $d_{F}(u)=f(u)+1$. By applying Lemma 4.6, the graph $G$ has a connected factor $H$ such that for each vertex $v$, $d_{H}(v)\in\{d_{F}(v),d_{F}(v)+1\}$, and also $d_{H}(u)=d_{F}(u)$. This implies that $H$ is a connected $\{f,f+1\}$-factor.

• $\Box$*

Corollary 4.9

.* Every $3$-tough $f(f+1)$-iso-tough graph $G$ has a connected $\{f,f+1\}$-factor, where $f$ is a positive integer-valued function on $V(G)$ with $f\geq 3$.*

**Proof. **

Apply Theorem 4.8 with \text{\varepsilon}=1.

• $\Box$*

5 Graphs with higher toughness

Our in this section is to provide another improvement for Theorem 1.3. Before doing so, let us refine Theorem 4.5 slightly for graphs with higher toughness.

Theorem 5.1

.* Let $G$ be a graph, let $f$ be a positive integer-valued function on $V(G)$, and let $a$ be a real number with $f\geq a>1$. For each vertex $v$, let \text{\varepsilon}_{0}(v)\in\{0,1\} such that \text{\varepsilon}_{0}(v)=1 if only if $f(v)$ and $a$ are integers with the same parity. Then $G$ has a near $f$-factor, if $G$ is \frac{1}{4(a-1)}((f(v)+a-1)^{2}-\text{\varepsilon}_{0}(v))-iso-tough and for all $S\subseteq V(G)$,*

[TABLE]

where $I_{*}(G\setminus S)$ is the set of center vertices of the star components of $G\setminus S$ in which for stars with one edge $xy$ the vertex $x$ is center whenever $f(x)\geq f(y)$.

**Proof. **

The proof presented here is inspired by the proof of Theorem 1 in [10]. Let $A$ and $B$ be two disjoint subsets of $V(G)$. For notational simplicity, we write $d(v)$ for $d_{G\setminus A}(v)$. Define $B_{0}=\{v\in B:d(v)<f(v)\}$. By Corollary 2.5, the graph $G[B_{0}]$ has an independent set $I_{0}$ such that

[TABLE]

For each vertex $v$, let t(v)=\frac{1}{4(a-1)}((f(v)+a-1)^{2}-\text{\varepsilon}_{0}(v)). Since $G$ is $t$-iso-tough, we have $\sum_{v\in I_{0}}t(v)\leq|A\cup N_{G}(I_{0})|\leq|A|+\sum_{v\in I_{0}}d(v)$. Since $d(v)$ is integer, we must have

[TABLE]

which implies that

[TABLE]

Therefore, Relations (1) and (2) can deduce that

[TABLE]

Let $I$ be a maximal independent set in $G[B]$ containing the vertices of $I_{0}$ so that $B\setminus I\subseteq N_{G}(I)$. Denote by $x_{1}$ the number of components $C$ of $G\setminus(A\cup B)$ such that $d_{G}(v,I)=1$ for each $v\in V(C)$. For every such a component $C$, select a vertex $z$ with $d_{G}(z,I)=1$. Define $Z$ to be the set of all selected vertices. Also, denote by $x_{2}$ the number of components $C$ of $G\setminus(A\cup B)$ such that $d_{G}(v,I)\geq 1$ for each $v\in V(C)$, and $d_{G}(u,I)\geq 2$ for at least one vertex $u\in V(C)$. Set $S=A\cup(N_{G}(I)\setminus Z)$. According to this definition, it is not difficult to show that

[TABLE]

and

[TABLE]

On the other hand, by the assumption,

[TABLE]

which implies that

[TABLE]

Since $d(v)\geq f(v)$ for each $v\in B\setminus(B_{0}\cup I)$, we must have

[TABLE]

Therefore, Relations (3) and (4) can conclude that

[TABLE]

Hence the assertion follows from Corollary 2.2.

• $\Box$*

The following corollary is an improved version of Theorem 1.3.

Corollary 5.2

.* A graph $G$ has a near $k$-factor, if for all $S\subseteq V(G)$, $iso(G\setminus S)\leq\frac{1}{k}|S|$, and*

[TABLE]

where $w_{*}(G\setminus S)$ denotes the number of star components of $G\setminus S$.

**Proof. **

Apply Theorem 5.1 with setting $f(v)=a=k$ when $k>1$. For the special case $k=1$, one can apply Theorem 1.1 directly.

• $\Box$*

Corollary 5.3

.([8])* Every $k$-tough graph $G$ of order at least $k+1$ has a near $k$-factor.*

**Proof. **

We may assume that $|V(G)|\geq k+2$. Let $S$ be a subset of $V(G)$. If $|S|<k$, then $\omega(G\setminus S)=1$ and also $iso(G\setminus S)=0$. Since $|V(G)|\geq k+2$, by the assumption, one can conclude that each vertex of $G$ contains at least $k+1$ neighbours. Hence $w_{*}(G\setminus S)=0$. If $|S|\geq k$, then we have $iso(G\setminus S)\leq|S|/k$ and $\omega(G\setminus S)+(k-1)\,w_{*}(G\setminus S)\leq k\,\omega(G\setminus S)\leq|S|$. Now, it is enough to apply Corollary 5.2.

• $\Box$*