Compatible Hamilton cycles in Dirac graphs

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on n ≥ 3 vertices with minimum degree at least n/2 is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we obtain the following strengthening of this result. Given a graph G = (V,E), an incompatibility system F over G is a family F = {Fv}v∈V such that for every v ∈ V , the set Fv is a family of unordered pairs Fv ⊆ {{e, e′} : e 6= e′ ∈ E, e∩ e′ = {v}}. An incompatibility system is ∆-bounded if for every vertex v and an edge e incident to v, there are at most ∆ pairs in Fv containing e. We say that a cycle C in G is compatible with F if every pair of incident edges e, e′ of C satisfies {e, e′} / ∈ Fv, where v = e∩ e′. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be viewed as a quantitative measure of robustness of graph properties. We prove that there is a constant μ > 0 such that for every μn-bounded incompatibility system F over a Dirac graph G, there exists a Hamilton cycle compatible with F . This settles in a very strong form a conjecture of Häggkvist from 1988.