On the maximum number of cliques in a graph embedded in a surface

Authors:
Vida Dujmović;GašPer Fijav;GwenaëL Joret;Thom Sulanke;David R. Wood
Affiliations:
School of Computer Science, Carleton University, Ottawa, Canada;Faculty of Computer and Information Science, University of Ljubljana, Ljubljana, Slovenia;Département d'Informatique, Université Libre de Bruxelles, Brussels, Belgium;Department of Physics, Indiana University, Bloomington, IN, USA;Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
Venue:
European Journal of Combinatorics
Year:
2011

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Abstract

This paper studies the following question: given a surface @S and an integer n, what is the maximum number of cliques in an n-vertex graph embeddable in @S? We characterise the extremal graphs for this question, and prove that the answer is between 8(n-@w)+2^@w and 8n+522^@w+o(2^@w), where @w is the maximum integer such that the complete graph K"@w embeds in @S. For the surfaces S"0, S"1, S"2, N"1, N"2, N"3 and N"4 we establish an exact answer.