An improved upper bound on the density of universal random graphs

Abstract

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/d n⌋ edges contain as subgraphs all graphs with n vertices and maximum degree at most d. © 2012 Springer-Verlag Berlin Heidelberg.