Universality for transversal powers of Hamilton cycles

TL;DR

This paper proves that under certain minimum degree conditions, any edge-colouring of the $k$th power of a Hamilton cycle can be embedded into a collection of graphs, extending previous results on Hamilton cycle conditions.

Contribution

It generalizes existing minimum degree conditions for Hamilton cycles to transversal powers of Hamilton cycles in edge-coloured graphs.

Findings

Establishes minimum degree threshold for embedding transversal powers of Hamilton cycles.

Extends prior work on Hamilton cycle conditions to more complex cycle powers.

Provides asymptotically optimal conditions for the existence of such embeddings.

Abstract

Let $k \ge 2$ and let $\bf G = \{G_1, \ldots, G_{m}\}$ be a collection of graphs on a common vertex set of cardinality $n$. We show that if each graph in $\bf G$ has minimum degree at least $(1-\frac{1}{2k} + o(1))n$, then for every edge-colouring $\chi$ of the $k$th power of a Hamilton cycle $C_n^k$ with $m$ colours, there is a copy of $C_n^k$ in $\bf G$ such that $e \in G_{\chi(e)}$ for every edge $e$ in $C_n^k$. This generalises a result of Bowtell, Morris, Pehova, and Staden, who provided asymptotically best possible minimum degree conditions for the Hamilton cycle.