Near-optimum Universal Graphs for Graphs with Bounded Degrees

• Authors: Noga Alon;Michael R. Capalbo;Yoshiharu Kohayakawa;Vojtech Rödl;Andrzej Rucinski;Endre Szemerédi
• Venue: APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
• Year: 2001

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(n2-2/k(log n)1+8/k) edges. This is optimal up to a small polylogarithmic factor, as Ω(n2-2/k) 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.