New isolated toughness condition for fractional -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.
