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 = {F_v}_v∈V such that for every v∈V, the set F_v is a family of unordered pairs F_v ⊆ {{e,e′}}: e ≠ e′ ∈ E,e ∩ e′ = {v}}. An incompatibility system is Δ-bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in F_v 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′}∉F_v, 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.