A bandwidth theorem for graph transversals

TL;DR

This paper establishes the asymptotically optimal minimum degree condition for finding spanning graph transversals with bounded degree and bandwidth in collections of graphs sharing the same vertex set, generalizing the Bandwidth theorem.

Contribution

It extends the Bandwidth theorem to graph transversals, providing the precise degree threshold for embedding graphs with bounded degree and bandwidth in multiple graphs.

Findings

Determined the asymptotic threshold for spanning transversals with bounded degree and bandwidth.

Generalized the Bandwidth theorem to the setting of graph transversals.

Provides a new criterion for embedding complex graphs in multiple graph collections.

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 $\lambda : E(H) \rightarrow [h]$ 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 B\"ottcher, Schacht and Taraz.