On the Minimum Degree of Minimally t -Tough, Claw-Free Graphs

Abstract

A graph is called minimally $t$-tough if the toughness of the graph is $t$ but the removal of any edge decreases the toughness. Katona and Varga conjectured that every minimally $t$-tough graph has a vertex of degree $\lceil 2t\rceil$. Matthews and Sumner proved that the toughness of any claw-free graph is always equal to half its connectivity, which implies that the toughness of a claw-free graph is always an integer or half of an integer. Katona et al. proved that this conjecture holds for minimally $t$-tough, claw-free graphs if $t\in \left(\frac{1}{2},1\right)$, and we proved before that this is also true if $t=\frac{3}{2}$. In this paper, we prove that every minimally $t$-tough, claw-free graph has a vertex of degree at most $\left(\frac{10t-5}{3}\right)$ for $t\ge 2$.