Universality for transversal Hamilton cycles in random graphs

TL;DR

The paper proves that an $n$-tuple of independent random graphs $(G_1,\dots,G_n)$ on the same vertex set is Hamilton-universal with high probability once $p\ge C\log n/n$ in $G(n,p)$. The authors adapt McDiarmid's coupling to transfer single-graph Hamiltonicity to rainbow color-patterns, and overcome a union-bound obstacle via an absorption strategy that peels off a small vertex set and employs Friedman-Pippenger embeddings to cover the remainder. The core contribution is a robust sparse-graph universality result for transversal Hamilton cycles, matching the deterministic threshold up to a constant factor and highlighting how randomness enables universal transversal structures beyond what is possible deterministically. This advances the understanding of rainbow subgraph universality in random settings and opens questions about rainbow universality thresholds and extensions to related spanning structures.

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 $χ: [n]\to[n]$ there exists a Hamilton cycle whose $i$-th edge comes from $G_{χ(i)}$. Bowtell, Morris, Pehova and Staden proved an analog of Dirac's theorem in this setting, namely that if $δ(G_i)\geq (1/2+o(1))n$ 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 $p\geq C\log n/n$, then $(G_1,\dots,G_n)$ is Hamilton-universal with high probability.