Perfect integer k-matching, k-factor-critical, and the spectral radius of graphs

A graph G is k-factor-critical if $G-S$ has a perfect matching for any subset S of $V\left(G\right)$ with $\left|S\right|=k$. An integer k-matching of G is a function $h:E\left(G\right)\to \{0,1,\dots ,k\}$ satisfying ${\sum }_{e\in \Gamma \left(v\right)}h\left(e\right)\le k$ for all $v\in V\left(G\right)$, where $\Gamma \left(v\right)$ is the set of edges incident with v. An integer k-matching h of G is called perfect if ${\sum }_{e\in E\left(G\right)}h\left(e\right)=k\left|V\right(G\left)\right|/2$. A graph G has the strong parity property if for every subset S of $V\left(G\right)$ with even size, G has a spanning subgraph F with minimum degree at least one such that $d_F\left(v\right)\equiv 1\left(\mathrm{mod}2\right)$ for all $v\in S$ and $d_F\left(u\right)\equiv 0\left(\mathrm{mod}2\right)$ for all $u\in V\left(G\right)﹨S$. In this paper, we provide edge number and spectral conditions for the k-factor-criticality, perfect integer k-matching and strong parity property of a graph, respectively.