Near-optimum Universal Graphs for Graphs with Bounded Degrees

Let H be a family of graphs. We say that G is H-universal if, for each H ∈ H, the graph G contains a subgraph isomorphic to H. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an H(k, n)-universal graph Γ (k, n) with O(n(log n)) edges. This is optimal up to a small polylogarithmic factor, as Ω(n) is a lower bound for the number of edges in any such graph. En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Γ (k, n), we prove, using a probabilistic argument, that Γ (k, n) is H(k, n)-universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties.