Maximum Clique

A maximum clique of a graph G is a clique (i.e., complete subgraph) of maximum possible size for G. Note that some authors refer to maximum cliques simply as "cliques." The size of the maximum clique is known as a graph's clique number, and the problem of finding the size of a clique for a given graph is an NP-complete problem (Skiena 1997).

A maximum clique in a given graph g can be found in the Wolfram Language using FindClique[g][[1]]. The command Sort[FindClique[g, Length /@ FindClique[g], All]] will find all maximum cliques.

A complete k-partite graph has maximum clique size k. The largest order-n graph which does not contain the complete graph K_p as a subgraph is called the Turán's graph T(n,p-1) (Gross and Yellen 2006, pp. 476-477; note the slightly nonstandard indexing T(n,p) by Skiena 1990, p. 218 and Pemmaraju and Skiena 2003, pp. 247-248).

See also

Caveman Graph, Clique, Clique Graph, Clique Number, Complete Graph, Induced Subgraph, Maximal Clique Party Problem, Perfect Graph, Ramsey Number, Turán's Theorem

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Cite this as:

Weisstein, Eric W. "Maximum Clique." From MathWorld--A Wolfram Web Resource.

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