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

For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that…

The toughness of a graph $G$ is defined as the minimum value of $|S|/c(G-S)$ over all cutsets $S$ of $G$ if $G$ is noncomplete, and is defined to be $\infty$ if $G$ is complete. For a real number $t$, we say that $G$ is $t$-tough if its…

The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if…

A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical…

A graph $G$ is called $k$-factor-critical if $G-S$ has a perfect matching for every $S\subseteq G$ with $|S|=k$. A connected graph $G$ is called $t$-connected if it has more than $t$ vertices and remains connected whenever fewer than $t$…

In this paper, we give a sufficient and necessary condition for a $k$-extendable graph to be $2k$-factor-critical when $k=\nu/4$, and prove some results on independence numbers in $n$-factor-critical graphs and $k\frac{1}{2}$-extendable…

For a graph $G$, let $odd(G)$ and $\omega(G)$ denote the number of odd components and the number of components of $G$, respectively. Then it is well-known that $G$ has a 1-factor if and only if $odd(G-S)\le |S|$ for all $S\subset V(G)$.…

Let $\mbox{odd}(G)$ and $i(G)$ denote the number of nontrivial odd components and the number of isolated vertices of a graph $G$, respectively. The $k$-Berge-Tutte-formula of a graph $G$ is defined as:…

A graph is equimatchable if each of its matchings is a subset of a maximum matching. It is known that any 2-connected equimatchable graph is either bipartite, or factor-critical, and that these two classes are disjoint. This paper provides…

Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=6$ for $\alpha\in[0,\frac{2}{3}]$ and $f(\alpha)=\frac{4}{1-\alpha}$ for $\alpha\in(\frac{2}{3},1)$. A graph $G$ is said to be…

For a set $\mathcal{H}$ of connected graphs, a spanning subgraph $H$ of $G$ is called an $\mathcal{H}$-factor of $G$ if each component of $H$ is isomorphic to an element of $\mathcal{H}$. A graph $G$ is called an $\mathcal{H}$-factor…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. $1$- and $2$-factor-critical graphs are the well-known factor-critical and…

Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest set $D$ of vertices of…

Let $G$ be a simple graph and let $n,m$ be two integers with $0<m<n$. We prove that $iso(G-S)\leq \frac{n}{m}|S|$ for every $S \subset V(G)$ if and only if $G$ has a $\{C_{2i+1},T \colon 1 \leq i < \frac{m}{n-m},…

Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest set $D$ of vertices of…

A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ is a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free graph has a 2-factor. The toughness condition of this result is…

A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with…

The $\{K_{1,1}, K_{1,2},C_m: m\geq3\}$-factor of a graph is a spanning subgraph whose each component is an element of $\{K_{1,1}, K_{1,2},C_m: m\geq3\}$. In this paper, through the graph spectral methods, we establish the lower bound of the…

Let $G$ be a graph. We denote by $e(G)$ and $\rho(G)$ the size and the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an even factor of $G$ if $d_F(v)\in\{2,4,6,\ldots\}$ for every $v\in V(G)$. Yan and Kano provided a…

Let $t$ be a positive integer, and let $G$ be a connected graph of order $n$ with $n\geq t+2$. A graph $G$ is said to be $\frac{1}{t}$-tough if $|S|\geq\frac{1}{t}c(G-S)$ for every subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is…