A bandwidth theorem for graph transversals

Abstract

Given a collection $\mathcal{G}=(G_1,\dots ,G_h)$ of graphs on the same vertex set $V$ of size $n$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $\lambda :E(H)\to [h]$ such that $e\in E(G_{\lambda (e)})$ for each $e\in E(H)$. The conditions on the minimum degree $\delta (\mathcal{G})={\min }_{i\in [h]}\{\delta (G_i)\}$ for finding a spanning $\mathcal{G}$-transversal isomorphic to a graph $H$ have been actively studied when $H$ is a Hamilton cycle, an $F$-factor, a spanning tree with maximum degree $o(n/\log n)$ and a power of a Hamilton cycle, etc. In this paper, we determined the asymptotically tight threshold on $\delta (\mathcal{G})$ for finding a $\mathcal{G}$-transversal isomorphic to $H$ when $H$ is a general $n$-vertex graph with bounded maximum degree and $o(n)$-bandwidth. This provides a transversal generalization of the celebrated Bandwidth theorem by Böttcher, Schacht and Taraz.