New isolated toughness condition for fractional $(g,f,n)$-critical graphs

Wei Gao¹ ; Weifan Wang²

¹ School of Information Science and Technology Yunnan Normal University Kunming 650500, China
² Department of Mathematics Zhejiang Normal University Jinhua 321004, China

Colloquium Mathematicum, Tome 147 (2017) no. 1, pp. 55-65.

Résumé

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.

Détail

DOI : 10.4064/cm6713-8-2016

Keywords: number isolated vertices graph isolated toughness defined infty complete operatorname min g s subseteq g s otherwise fractional critical graph bn vardelta where positive integers vardelta b a furthermore isolated toughness condition fractional critical graphs given