Spectral conditions for graphs having all (fractional) $[a,b]$-factors

Abstract

Let $a\le b$ be two positive integers. We say that a graph $G$ has all $[a,b]$-factors if it has an $h$-factor for every function $h:V(G)\to {\mathbb{Z}}^{+}$ such that $a\le h(v)\le b$ for all $v\in V(G)$ and ${\sum }_{v\in V(G)}h(v)\equiv 0(\mathrm{mod}2)$, and has all fractional $[a,b]$-factors if it has a fractional $p$-factor for every $p:V(G)\to {\mathbb{Z}}^{+}$ such that $a\le p(v)\le b$ for all $v\in V(G)$. In this paper, we provide tight spectral radius conditions for graphs having all $[a,b]$-factors ($3\le a<b$) and all fractional $[a,b]$-factors ($1\le a<b$), respectively.

arXiv comment15 pages