Non-hamiltonian Graphs with Given Toughness

Abstract

In 1973, Chvátal introduced the concept of toughness τ of a graph and conjectured that there exists a finite constant t0 such that every t0-tough graph (that is τ ≥ t0) is hamiltonian. To solve this challenging problem, all efforts are directed towards constructing non-hamiltonian graphs with toughness as large as possible. The last result in this direction is due to Bauer, Broersma and Veldman, which states that for each positive ϵ, there exists a non-hamiltonian graph with 9/4 - ϵ ≤ τ < 9/4. The following related broad-scale problem, reminding the well-known pancyclicity or hypohamiltonicity, arises naturally: whedher there exists a non-hamiltonian graph with a given toughness. We conjecture that if there exist a non-hamiltonian t-tough graph then for each rational number a with 0 < a ≤ t there exists a non-hamiltonian graph whose toughness is exactly a. In this paper we prove this conjecture for t = 9/4 - ϵ by using a number of additional modified building blocks to construct the required graphs.