Compatible Hamilton cycles in Dirac graphs

TL;DR

This paper proves that Dirac graphs, which have high minimum degree, still contain Hamilton cycles compatible with certain bounded incompatibility systems, strengthening classical Hamiltonicity results.

Contribution

It establishes that even with bounded incompatibility constraints, Dirac graphs retain Hamiltonicity, confirming a longstanding conjecture by H"{a}ggkvist.

Findings

Existence of compatible Hamilton cycles in Dirac graphs under bounded incompatibility

Strengthening of Dirac's theorem with robustness considerations

Resolution of H"{a}ggkvist's conjecture from 1988

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 $n\ge 3$ 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 {\em incompatibility system} $\mathcal{F}$ over $G$ is a family $\mathcal{F}=\{F_v\}_{v\in V}$ such that for every $v\in V$, the set $F_v$ is a set of unordered pairs $F_v \subseteq \{\{e,e'\}: e\ne e'\in E, e\cap e'=\{v\}\}$. An incompatibility system is {\em $\Delta$-bounded} if for every vertex $v$ and an edge $e$ incident to $v$, there are at most $\Delta$ pairs in $F_v$ containing $e$. We say that a cycle $C$ in $G$ is {\em compatible} with $\mathcal{F}$ if every pair of incident edges $e,e'$ of $C$ satisfies $\{e,e'\} \notin F_v$, where $v=e\cap e'$. 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 $\mu>0$ such that for every $\mu n$-bounded incompatibility system $\mathcal{F}$ over a Dirac graph $G$, there exists a Hamilton cycle compatible with $\mathcal{F}$. This settles in a very strong form, a conjecture of H\"{a}ggkvist from 1988.