Robust Hamiltonicity in families of Dirac graphs

TL;DR

This paper investigates the existence and quantity of Hamilton cycle transversals in collections of Dirac graphs, establishing thresholds, counting results, and edge-disjoint configurations, thereby extending classical Hamilton cycle results.

Contribution

It determines the threshold for Hamilton cycle transversals in random subgraphs of Dirac graphs and proves the existence of many such transversals and edge-disjoint collections, generalizing classical results.

Findings

At least (cn)^{2n} Hamilton cycle transversals in collections of Dirac graphs.

Thresholds for Hamilton cycle transversals in random Dirac subgraphs.

Existence of n/2 edge-disjoint Hamilton cycle transversals in large collections.

Abstract

A graph is called Dirac if its minimum degree is at least half of the number of vertices in it. Joos and Kim showed that every collection $\mathbb{G}=\{G_1,\ldots,G_n\}$ of Dirac graphs on the same vertex set $V$ of size $n$ contains a Hamilton cycle transversal, i.e., a Hamilton cycle $H$ on $V$ with a bijection $\phi:E(H)\rightarrow [n]$ such that $e\in G_{\phi(e)}$ for every $e\in E(H)$. In this paper, we determine up to a multiplicative constant, the threshold for the existence of a Hamilton cycle transversal in a collection of random subgraphs of Dirac graphs in various settings. Our proofs rely on constructing a spread measure on the set of Hamilton cycle transversals of a family of Dirac graphs. As a corollary, we obtain that every collection of $n$ Dirac graphs on $n$ vertices contains at least $(cn)^{2n}$ different Hamilton cycle transversals $(H,\phi)$ for some absolute constant $c>0$. This is optimal up to the constant $c$. Finally, we show that if $n$ is sufficiently large, then every such collection spans $n/2$ pairwise edge-disjoint Hamilton cycle transversals, and this is best possible. These statements generalize classical counting results of Hamilton cycles in a single Dirac graph.