The minimum degree of minimally $t$-tough graphs

TL;DR

This paper investigates the minimum degree bounds of minimally t-tough graphs, providing new upper bounds based on girth and toughness, and confirming conjectures for specific graph classes.

Contribution

It establishes new bounds on the minimum degree of minimally t-tough graphs depending on girth and toughness, advancing understanding of their structural properties.

Findings

Minimally 1-tough graphs with girth ≥ 5 have minimum degree ≤ ⌊n/(g+1)⌋+g-1.

Minimally 1-tough graphs with girth 4 have minimum degree ≤ (n+6)/4.

Minimum degree of minimally 3/2-tough claw-free graphs is 3.

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 $ \lceil 2t\rceil $ and gave some upper bounds on the minimum degree of the minimally $ t $-tough graphs in \cite{Katona, Gyula}. In this paper, we show that a minimally 1-tough graph $ G $ with girth $ g\geq 5 $ has minimum degree at most $ \lfloor\frac{n}{g+1}\rfloor+g-1$, and a minimally $ 1 $-tough graph with girth $ 4 $ has minimum degree at most $ \frac{n+6}{4}$. We also prove that the minimum degree of minimally $\frac{3}2$-tough claw-free graphs is $ 3 $.