Compatible Hamilton cycles in Dirac graphs

Michael Krivelevich, Choongbum Lee, Benny Sudakov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 ≠ 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 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.

Original languageEnglish
Pages (from-to)697-732
Number of pages36
Issue number4
StatePublished - 1 Aug 2017


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