Isolated Toughness and Existence of $[a,b]$-Factors in Graphs

Abstract

For a graph $G$ with vertex set $V(G)$ and edge set $E(G)$, let $i(G)$ be the number of isolated vertices in $G$. The \emph{isolated toughness} of $G$ is defined as

$$I(G)=min\{\frac{|S|}{i(G-S)}\mid S\subseteq V(G),i(G-S)\ge 2\},$$

if $G$ is not complete; and $I(K_n)=n-1$. In this paper, we investigate the existence of $[a,b]$-factors in terms of this graph invariant. We proved that if $G$ is a graph with $\delta (G)\ge a$ and $I(G)\ge a$, then $G$ has a fractional $a$-factor. Moreover, if $\delta (G)\ge a$, $I(G)>(a-1)+\frac{a-1}{b}$, and $G-S$ has no $(a-1)$-regular component for any subset $S$ of $V(G)$, then $G$ has an $[a,b]$-factor. The latter result is a generalization of Katerinis’ well-known theorem about $[a,b]$-factors (P. Katerinis, Toughness of graphs and the existence of factors, \emph{Discrete Math}. 80(1990), 81-92).