Compatible Hamilton cycles in Dirac graphs

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),{v}}. An incompatibility system is ∆-bounded if for every vertex v and an edge e incident to v, there are at most ∆ 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 satisfiesThis 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.

Rainbow matchings in Dirac bipartite graphs

“…The following result establishes a dichotomy between these classes. Similar ideas have already appeared in previous work on Dirac graphs . The proof follows the lines of the previous approaches and we include it here for the sake of completeness.…”

“…A natural candidate is to embed Hamiltonian cycles in Dirac graphs. Krivelevich coworkers [15] proved the existence of  -conflict-free Hamiltonian cycles in Dirac graphs, provided that the conflicts in  are local. Their proof is substantially different from ours and relies on Pósa rotations.…”

Robustness of graph properties

“…Can we find a compatible Hamilton cycle for every ∆-bounded system, when ∆ is linear in n? The following theorem, proved by Krivelevich, Lee and the author [67], answers these questions.…”

“…The proof in [67] provides the existence of a positive constant µ of approximately 10 −16 (although no serious attempt was made to optimize it). On the other hand, the following variant of a construction of Bollobás and Erdős [15] shows that µ is at most 1 4 .…”

Compatible Hamilton cycles in random graphs

“…Further study of how various extremal results can be strengthened using this notion appears to be a promising direction of research. For example in a companion paper [23], we show that there exists a constant μ > 0 such that for any μn-bounded system F over a graph G on n vertices with minimum degree at least n/2, there is a compatible Hamilton cycle in G. This establishes in a very strong sense an old conjecture of Häggkvist from 1988.…”