How Many Conjectures Can You Stand? A Survey

Abstract

We survey results and open problems in hamiltonian graph theory centered around two conjectures of the 1980s that are still open: every 4-connected claw-free graph (line graph) is hamiltonian. These conjectures have lead to a wealth of interesting concepts, techniques, results and equivalent conjectures.

The Nash-Williams Conjecture and the Dominating Cycle Conjecture

“…If L(G) has a hamiltonian cycle, then G has a DC by Th. 5 in [2] (see also [3]) and we are finished. For the formulation and proof of the next results we use the following definition.…”

“…For used terminology which is not defined here we refer to [1,2]. A dominating cycle (DC) of a graph G is a cycle which contains at least one endvertex of every edge of G. Let v ∈ V (G) then E v denotes the set of edges incident with v. A closed trail is a closed walk in which all the edges are distinct.…”

A Relationship Between Thomassen's Conjecture and Bondy's Conjecture

“…Together with Proposition 1.3, we see the following situation:Conjecture 1.1 = ======= =⇒ Conjecture 1.4.However, we do not know about the converse of these two implications. Indeed, as mentioned in section 1, the converse of Proposition 1.3 appeared in[6] as an open problem. In addition to that, we left an open problem on the converse of Theorem 1.5. gives a corollary concerning Conjecture 1.4.…”

Hamilton cycles in almost distance-hereditary graphs

“…Many results about the existence of Hamilton cycles in claw-free graphs have been obtained. For surveys on Matthews and Sumner's conjecture and on claw-free graphs, we refer the reader to [5] and [9], respectively.…”