Size and spectral conditions for a graph with given minimum degree to be $k$-$d$-critical

Abstract

A $k$-matching in a graph $G$ is defined as a function $f:E(G)\to \{0,1,\dots ,k\}$ satisfying ${\sum }_{e\in E_G(v)}f(e)$ $\le k$ for each vertex $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident to $v$ in $G$. For $1\le d\le k$ and $d\equiv |V(G)|(\mathrm{mod}2)$, if for any $v\in V(G)$, there exists a $k$-matching $f$ such that ${\sum }_{e\in E_G(v)}f(e)=k-d$ and ${\sum }_{e\in E_G(u)}f(e)=k\text{ for any }u\in V(G)-\{v\}$, then $G$ is $k$-$d$-critical. A graph $G$ of odd order (resp. even order) is generalized factor-critical (resp. generalized bicritical) if the empty set is the unique set attaining the maximum value in $k$-Berge-Tutte-formula of $G$. In this paper, we provide sharp sufficient conditions in terms of size or spectral radius respectively for a graph $G$ to be $k$-$d$-critical, generalized factor-critical and generalized bicritical with minimum degree.