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arXiv · 1607.03234

Optimal induced universal graphs for bounded-degree graphs

Noga Alon·Rajko Nenadov
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Abstract

We show that for any constant $Δ\ge 2$, there exists a graph $G$ with $O(n^{Δ/ 2})$ vertices which contains every $n$-vertex graph with maximum degree $Δ$ as an induced subgraph. For odd $Δ$ 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 $Ω(n^{Δ/2})$ 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 $Δ$, an induced subgraph of $G$ isomorphic to $H$.

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Noga Alon, Rajko Nenadov. 2016-07-12. Optimal induced universal graphs for bounded-degree graphs. https://doi.org/10.1017/s0305004117000706

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