A note on classes of subgraphs of locally finite graphs

TL;DR

This paper explores the minimal size of locally finite graphs containing all graphs from a certain class, providing conditions for their existence and addressing whether such graphs can have bounded maximum degree.

Contribution

It establishes necessary and sufficient conditions for the existence of a connected, locally finite graph containing all members of a given class, and answers a question about bounded degree graphs.

Findings

Such a universal graph exists under certain conditions.

For the class of graphs with bounded degree, the universal graph cannot have bounded maximum degree.

The paper resolves a recent open question in the field.

Abstract

We investigate the question how `small' a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a connected, locally finite graph $H$ containing all elements of a graph class $\mathcal G$. These conditions imply that such a graph $H$ exists for the class $\mathcal G_d$ consisting of all graphs with maximum degree $<d$ which raises the question whether in this case $H$ can be chosen to have bounded maximum degree. We show that this is not the case, thereby answering a question recently posed by Huynh et al.