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arXiv · 2510.18163

Universality for transversal powers of Hamilton cycles

Emily Heath·Joseph Hyde·Natasha Morrison·Shannon Ogden
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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.

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Emily Heath, Joseph Hyde, Natasha Morrison, Shannon Ogden. 2025-10-20. Universality for transversal powers of Hamilton cycles. https://arxiv.org/abs/2510.18163

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