On the minimum degree of minimally 1-tough, triangle-free graphs and minimally 3/2-tough, claw-free graphs

Abstract

A graph G is minimally t-tough if the toughness of G is t and deletion of any edge from G decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally t-tough graph is ⌈2t⌉ and gave some upper bounds on the minimum degree of the minimally t-tough graphs in [5,6]. In this paper, we show that a minimally 1-tough graph G with girth g≥5 has minimum degree at most ⌊ng+1⌋+g−1, and a minimally 1-tough graph with girth 4 has minimum degree at most n+64. We also prove that the minimum degree of minimally 32-tough, claw-free graphs is 3.