Universality for transversal Hamilton cycles in random graphs

Abstract

A tuple $(G_1,\dots ,G_n)$ of graphs on the same vertex set of size $n$ is said to be Hamilton-universal if for every map $\ddot{I}‡:[n]\to [n]$ there exists a Hamilton cycle whose $i$-th edge comes from $G_{\ddot{I}‡(i)}$. Bowtell, Morris, Pehova and Staden proved an analog of Dirac's theorem in this setting, namely that if $\hat{I}´(G_i)\ge (1/2+o(1))n$ then $(G_1,\dots ,G_n)$ is Hamilton-universal. Combining McDiarmid's coupling and a colorful version of the Friedman-Pippenger tree embedding technique, we establish a similar result in the setting of sparse random graphs, showing that there exists $C$ such that if the $G_i$ are independent random graphs sampled from $G(n,p)$, where $p\ge C\log n/n$, then $(G_1,\dots ,G_n)$ is Hamilton-universal with high probability.

16 pages