Universality for transversal Hamilton cycles

TL;DR

This paper proves that collections of dense graphs on the same vertex set contain all possible Hamilton cycle patterns, demonstrating a form of universality in such graph collections.

Contribution

It establishes that dense graph collections are universal for all Hamilton cycle patterns, extending previous results to a broader setting.

Findings

Contains every Hamilton cycle pattern in dense graph collections

Shows universality for Hamilton cycles in graph collections

Extends known results to more general graph collections

Abstract

Let $\mathbf{G}=\{G_1, \ldots, G_m\}$ be a graph collection on a common vertex set $V$ of size $n$ such that $\delta(G_i) \geq (1+o(1))n/2$ for every $i \in [m]$. We show that $\mathbf{G}$ contains every Hamilton cycle pattern. That is, for every map $\chi: [n] \to [m]$ there is a Hamilton cycle whose $i$-th edge lies in $G_{\chi(i)}$.