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Universality for transversal Hamilton cycles
Candida Bowtell, Patrick Morris, Yanitsa Pehova, Katherine Stade · 2023-10-06 · via math updates on arXiv.org

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