An Improved Upper Bound on the Density of Universal Random Graphs

We give a polynomial time randomized algorithm that, on receiving as input a pair (H,G) of n-vertex graphs, searches for an embedding of H into G. If H has bounded maximum degree and G is suitably dense and pseudorandom, then the algorithm succeeds with high probability. Our algorithm proves that, for every integer d≥3 and suitable constant C=Cd, as n→∞, asymptotically almost all graphs with n vertices and ⌊Cn2-1/d log1/dn⌋ edges contain as subgraphs all graphs with n vertices and maximum degree at most d.

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