Nonhamiltonian Graphs with Given Toughness

TL;DR

This paper proves that for any rational toughness value between 0 and 9/4, there exists a nonhamiltonian graph with that toughness, extending previous results to a continuous range.

Contribution

It constructs nonhamiltonian graphs with any rational toughness value in the interval (0, 9/4), filling a gap in the understanding of graph toughness and Hamiltonicity.

Findings

Existence of nonhamiltonian graphs for all rational toughness in (0, 9/4)

Extension of previous toughness bounds for nonhamiltonian graphs

Provides a complete characterization of possible toughness values below 9/4

Abstract

In 1973, Chv\'{a}tal introduced the concept of toughness $\tau$ of a graph and constructed an infinite class of nonhamiltonian graphs with $\tau=3/2$. Later Thomassen found nonhamiltonian graphs with $\tau>3/2$, and Enomoto et al. constructed nonhamiltonian graphs with $\tau=2-\epsilon$ for each positive $\epsilon$. The last result in this direction is due to Bauer, Broersma and Veldman, which states that for each positive $\epsilon$, there exists a nonhamiltonian graph with $\tau\ge 9/4-\epsilon$. In this paper we prove that for each rational number $t$ with $0<t<9/4$, there exists a nonhamiltonian graph with $\tau=t$.