Abstract
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.
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Research supported in part by USA-Israel BSF Grant 2010115 and by grant 912/12 from the Israel Science Foundation.
Research supported in part by NSF Grant DMS-1362326.
Research supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant.
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Krivelevich, M., Lee, C. & Sudakov, B. Compatible Hamilton cycles in Dirac graphs. Combinatorica 37, 697–732 (2017). https://doi.org/10.1007/s00493-016-3328-7
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DOI: https://doi.org/10.1007/s00493-016-3328-7