Universality for transversal Hamilton cycles

Abstract

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 $G_{χ(i)}$.