Graphs and Combinatorics

, Volume 23, Issue 3, pp 337–352

On the Maximum Number of Cliques in a Graph

Article

DOI: 10.1007/s00373-007-0738-8

Cite this article as:
Wood, D.R. Graphs and Combinatorics (2007) 23: 337. doi:10.1007/s00373-007-0738-8

Abstract

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K5-minor or no K3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2).

Keywords

extremal graph theory Turán’s Theorem clique Complete subgraph Degeneracy Graph minor Planar graph K5-minor K3, 3-minor 

Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

  1. 1.Departament de Matemática Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain