Some properties of (a,b,k)-critical graphs

Abstract

Let $a,b$ and $k$ be nonnegative integers with $1\leq a\leq b$, and let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$.
Then a spanning subgraph $F$ of $G$ is called an $[a,b]$-factor if $a\leq d_F(v)\leq b$ for any $v\in V(G)$. A graph $G$ is said to be
$(a,b,k)$-critical if $G-D$ contains an $[a,b]$-factor for each subset $D$ of $k$ elements of $V(G)$. We use $|E(G)|$ and $\rho(G)$ to denote the size and spectral radius, respectively. In this paper, we establish a lower bound on the size and spectral radius of a graph $G$ to ensure that $G$ is $(a,b,k)$-critical, respectively.