On hamiltonian properties of 2‐tough graphs

Abstract

A well-known conjecture in hamiltonian graph theory states that every 2-tough graph is hamiltonian. We give some equivalent conjectures, e.g., the conjecture that every 2-tough graph is hamiltonianconnected.

How Many Conjectures Can You Stand? A Survey

“…If G is connected this H is unique, except for G = K 3 : then H can be K 3 or K 1,3 (this is different for multigraphs, where we could also have three parallel edges, or two parallel edges and one additional incident edge; and there are other pairs of connected multigraphs with isomorphic line graphs). If we restrict ourselves to simple graphs and take K 1,3 in this exceptional case, we can talk of a unique graph H as the root graph of the connected line graph G isomorphic to L(H ). What is the counterpart in H of a Hamilton cycle in G?…”

“…Similar techniques were introduced and applied in [3] to obtain equivalent versions of the 2-tough conjecture, and in [4] successfully applied with suitable small gadgets to obtain counterexamples to the 2-tough conjecture. Although the 2-tough conjecture restricted to claw-free graphs is equivalent to Conjecture 1, it is beyond the scope of this survey.…”

More Progress on Tough Graphs - The Y2K Report

“…In [3] a construction of a nontraceable graph from non-hamiltonian-connected building blocks was used to show that Chvátal's conjecture on the hamiltonicity of 2-tough graphs is equivalent to several other statements, some seemingly weaker, some seemingly stronger. This construction was inspired by examples of graphs of high toughness without 2-factors occurring in [9].…”

“…We now give a brief outline of the construction of these counterexamples. The proof of the following theorem occurs in [4] and almost literally also in [3].…”

“…For many years, however, the focus was on determining whether all 2-tough graphs are hamiltonian. One reason for this is that if all 2-tough graphs are hamiltonian, a number of important consequences [3] would follow. In addition, the results below (for k = 2) seemed to indicate that two might be the threshold for toughness that would imply hamiltonicity.…”