Nonhamiltonian Graphs with Given Toughness

Abstract

In 1973, Chvátal introduced the concept of toughness $τ$ of a graph and constructed an infinite class of nonhamiltonian graphs with $τ=3/2$. Later Thomassen found nonhamiltonian graphs with $τ>3/2$, and Enomoto et al. constructed nonhamiltonian graphs with $τ=2-ε$ for each positive $ε$. The last result in this direction is due to Bauer, Broersma and Veldman, which states that for each positive $ε$, there exists a nonhamiltonian graph with $τ\ge 9/4-ε$. In this paper we prove that for each rational number $t$ with $0<t<9/4$, there exists a nonhamiltonian graph with $τ=t$.