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

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 inverted right perpendicular2tinverted left perpendicular and gave some upper bounds on the minimum degree graph G with girth g >= 5 has minimum degree at most left perpendicular n/g+1 right perpendicular + g - 1 and a minimally 1-tough graph with girth 4has minimum degree at most n+6/4. We also prove that the minimum degree of minimally 3/2-tough, claw-free graphs is 3. (c) 2023 Elsevier B.V. All rights reserved.