Optimal induced universal graphs for bounded-degree graphs

TL;DR

This paper constructs explicit graphs with optimal size that contain all bounded-degree graphs as induced subgraphs, improving previous bounds for odd degrees and providing an efficient method to find these subgraphs.

Contribution

It introduces a new explicit construction of universal graphs for bounded-degree graphs with optimal size bounds and an efficient algorithm for embedding any such graph.

Findings

Constructs universal graphs with $O(n^{rac{ ext{max degree}}{2}})$ vertices.

Improves bounds for odd maximum degrees over previous work.

Provides a deterministic algorithm for finding induced subgraphs.

Abstract

We show that for any constant $\Delta \ge 2$, there exists a graph $G$ with $O(n^{\Delta / 2})$ vertices which contains every $n$-vertex graph with maximum degree $\Delta$ as an induced subgraph. For odd $\Delta$ this significantly improves the best-known earlier bound of Esperet et al. and is optimal up to a constant factor, as it is known that any such graph must have at least $\Omega(n^{\Delta/2})$ vertices. Our proof builds on the approach of Alon and Capalbo (SODA 2008) together with several additional ingredients. The construction of $G$ is explicit and is based on an appropriately defined composition of high-girth expander graphs. The proof also provides an efficient deterministic procedure for finding, for any given input graph $H$ on $n$ vertices with maximum degree at most $\Delta$, an induced subgraph of $G$ isomorphic to $H$.