Related papers: Tight Toughness and Isolated Toughness for $\{K_2,…

The $\{K_2,C_n\}$-factor of a graph is a spanning subgraph whose each component is either $K_2$ or $C_n$. In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the…

For a graph $G = (V(G), E(G))$, let $i(G)$ be the number of isolated vertices in $G$. The {\it isolated toughness} of $G$ is defined as $I(G) = min\{|S|/i(G-S) : S\subseteq V(G), i(G-S)\geq 2\}$ if $G$ is not complete; $I(G)=|V(G)|-1$…

A spanning subgraph $F$ of $G$ is called a path factor if every component of $F$ is a path of order at least 2. Let $k\geq2$ be an integer. A $P_{\geq k}$-factor of $G$ means a path factor in which every component has at least $k$ vertices.…

Let $G$ be a connected graph with $n$ vertices. The isolated toughness of $G$, denoted by $I(G)$, is defined by $I(G)=\min\left\{\frac{|S|}{i(G-S)}:S\subseteq V(G) \ \mbox{and} \ i(G-S)\geq2\right\}$ if $G$ is not complete, or…

A spanning subgraph $H$ of a graph $G$ is called a $P_{\geq k}$-factor of $G$ if every component of $H$ is isomorphic to a path of order at least $k$, where $k\geq2$ is an integer. A graph $G$ is called a $(P_{\geq k},l)$-factor critical…

A graph is said to be $K_{1,r}$-free if it does not contain an induced subgraph isomorphic to $K_{1,r}$. An $\mathcal{F}$-factor is a spanning subgraph $H$ such that each connected component of $H$ is isomorphic to some graph in…

Let $G$ denote a graph and $k\geq2$ be an integer. A $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$-factor of $G$ is a spanning subgraph, whose every connected component is isomorphic to an element of…

Let $G$ be a graph of order $n$, and let $a$ and $b$ be two integers with $1\leq a\leq b$. Let $h: E(G)\rightarrow [0,1]$ be a function. If $a\leq\sum_{e\ni x}h(e)\leq b$ holds for any $x\in V(G)$, then we call $G[F_h]$ a fractional…

A graph is $t$-tough if the deletion of any set of, say, $m$ vertices from the graph leaves a graph with at most $\frac{m}{t}$ components. In 1973, Chv\'{a}tal suggested the problem of relating toughness to factors in graphs. In 1985,…

For a graph $G = (V, E)$, a {\em fractional $[a, b]$-factor} is a real valued function $h:E(G)\to [0,1]$ that satisfies $a \le ~ \sum_{e\in E_G(v)} h(e) ~ \le b$ for all $ v\in V(G)$, where $a$ and $b$ are real numbers and $E_G(v)$ denotes…

Let $\mathcal{A}$ be a set of connected graphs. Then a spanning subgraph $A$ of $G$ is called an $\mathcal{A}$-factor if each component of $A$ is isomorphic to some member of $\mathcal{A}$. Especially, when every graph in $\mathcal{A}$ is a…

A spanning subgraph $F$ of a graph $G$ is called a $[1,b]$-odd factor if $b\equiv1$ (mod 2) and $d_F(v)\in\{1,3,\ldots,b\}$ for every $v\in V(G)$. A graph $G$ of order $n\geq k+2$ is $k$-critical with respect to $[1,b]$-odd factor if for…

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A graph is $ID$-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. For two positive…

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a…

Let $G$ be a graph and $h: E(G)\rightarrow [0,1]$ be a function. For any two positive integers $a$ and $b$ with $a\leq b$, a fractional $[a,b]$-factor of $G$ with the indicator function $h$ is a spanning subgraph with vertex set $V(G)$ and…

A spanning subgraph of a graph G is called a [0,2]-factor of G, if for . is a union of some disjoint cycles, paths and isolate vertices, that span the graph G. It is easy to get a [0,2]-factor of G and there would be many of [0,2]-factors…

In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under…

A graph $G$ is $k$-factor-critical if $G-S$ has a perfect matching for every subset $S \subseteq V(G)$ with $|S|=k$. A spanning subgraph $H$ of $G$ is called a $[1,b]$-odd factor if $b \equiv 1 \pmod{2}$ and $d_{H}(v) \in\left\lbrace 1, 3,…

Let $G$ be a graph with vertex set $V(G)$. Let $n$ and $k$ be non-negative integers such that $n + 2k \leq |V(G)| - 2$ and $|V(G)| - n$ is even. If when deleting any $n$ vertices of $G$ the remaining subgraph contains a matching of $k$…

A near-factor of a finite simple graph $G$ is a matching that saturates all vertices except one. A graph $G$ is said to be near-factor-critical if the deletion of any vertex from $G$ results in a subgraph that has a near-factor. We prove…