Toughness and Vertex Degrees

TL;DR

This paper investigates degree-based conditions that ensure a graph's toughness, providing optimal theorems for different toughness levels and revealing the complexity of such conditions as toughness decreases.

Contribution

It presents the best monotone theorems for graph toughness based on vertex degrees, including a simple criterion for toughness less than one and complexity results for higher toughness levels.

Findings

Best monotone theorem for $t \\ge 1$ established.

Superpolynomial growth of conditions needed for $t=1/k$ with $k \\ge 1$.

Simple degree-based criterion for $t<1$ provided.

Abstract

We study theorems giving sufficient conditions on the vertex degrees of a graph $G$ to guarantee $G$ is $t$-tough. We first give a best monotone theorem when $t\ge1$, but then show that for any integer $k\ge1$, a best monotone theorem for $t=\frac1k\le 1$ requires at least $f(k)\cdot|V(G)|$ nonredundant conditions, where $f(k)$ grows superpolynomially as $k\rightarrow\infty$. When $t<1$, we give an additional, simple theorem for $G$ to be $t$-tough, in terms of its vertex degrees.