A problem of Erdos on the minimum number of k-cliques

Shagnik Das; Hao Huang; Jie Ma; Humberto Naves; Benny Sudakov

doi:10.1016/j.jctb.2013.02.003

A problem of Erdos on the minimum number of k-cliques

Shagnik Das, Hao Huang, Jie Ma, Humberto Naves, Benny Sudakov

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22 Scopus citations

Abstract

Fifty years ago Erdos asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l-1 complete graphs of size nl-1. This conjecture was disproved by Nikiforov, who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n2.In this paper we solve Erdos' problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdos' conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).

Original language	English (US)
Pages (from-to)	344-373
Number of pages	30
Journal	Journal of Combinatorial Theory. Series B
Volume	103
Issue number	3
DOIs	https://doi.org/10.1016/j.jctb.2013.02.003
State	Published - May 2013
Externally published	Yes

Bibliographical note

Funding Information:
E-mail addresses: shagnik@ucla.edu (S. Das), huanghao@math.ucla.edu (H. Huang), jiema@math.ucla.edu (J. Ma), hnaves@math.ucla.edu (H. Naves), bsudakov@math.ucla.edu (B. Sudakov). 1 Research supported by a UC Dissertation Year Fellowship. 2 Research supported in part by NSF grant DMS-1101185, by AFOSR MURI grant FA9550-10-1-0569 and by a USA–Israeli BSF grant.

Keywords

Clique density
Erdos' conjecture
Flag algebras

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abstract = "Fifty years ago Erdos asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l-1 complete graphs of size nl-1. This conjecture was disproved by Nikiforov, who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n2.In this paper we solve Erdos' problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdos' conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).",

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N2 - Fifty years ago Erdos asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l-1 complete graphs of size nl-1. This conjecture was disproved by Nikiforov, who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n2.In this paper we solve Erdos' problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdos' conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).

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A problem of Erdos on the minimum number of k-cliques

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Bibliographical note

Keywords

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