An improved upper bound on the density of universal random graphs

Authors:
Domingos Dellamonica;Yoshiharu Kohayakawa;Vojtěch Rödl;Andrzej Ruciński
Affiliations:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA;Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA and Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil;Department of Mathematics and Computer Science, Emory University, Atlanta, GA;Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA and Department of Discrete Mathematics, Adam Mickiewicz University, Poznań, Poland
Venue:
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Year:
2012

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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/dn⌋ edges contain as subgraphs all graphs with n vertices and maximum degree at most d.