New isolated toughness condition for fractional $(g,f,n)$-critical graphs
Volume 147 / 2017
Colloquium Mathematicum 147 (2017), 55-65
MSC: Primary 05C70.
DOI: 10.4064/cm6713-8-2016
Published online: 8 December 2016
Abstract
Let $i(G)$ be the number of isolated vertices in a graph $G$. The isolated toughness of $G$ is defined as $I(G)=\infty $ if $G$ is complete, and $I(G)=\operatorname{min}\{|S|/i(G-S) : S\subseteq V(G),\, i(G-S)\ge 2\}$ otherwise. We show that $G$ is a fractional $(g,f,n)$-critical graph if $I(G)\ge (b^{2}+bn-\varDelta )/{a}$, where $a, b$ are positive integers, $1\le a\le b$, $b\ge 2$, and $\varDelta =b-a$. Furthermore, a new isolated toughness condition for fractional $(a,b,n)$-critical graphs is given.
