Sparse universal graphs for bounded-degree graphs

Noga Alon*, Michael Capalbo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


Let ℋ be a family of graphs. A graph T is ℋ-universal if it contains a copy of each H ∈ ℋ as a subgraph. Let ℋ(k, n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an ℋ(k, n)-universal graph T with Ok(n2-2/k log4/k n) edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n 2-2/k) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders.

Original languageEnglish
Pages (from-to)123-133
Number of pages11
JournalRandom Structures and Algorithms
Issue number2
StatePublished - Sep 2007


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