A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. A maximum clique (i.e., clique of largest size in a given graph) is therefore always maximal, but the converse does not hold. Maximal cliques are important in graph theoretic applications, including graph coloring, fractional graph coloring, and computation of other graph properties such as intersection number.
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