In fact, sufficiently large graphs with no large cliques have large independent sets, a theme that is explored in Ramsey theory.
If a graph has sufficiently many edges, it must contain a large clique.
Graphs with large cliques have high chromatic number, but the opposite is not true.
Another local property that leads to high chromatic number is the presence of a large clique.
Because a rook's graph is perfect, the number of colors needed in any coloring of the graph is just the size of its largest clique.
If a claw-free graph is not perfect, it is NP-hard to find to find its largest clique.
A graph is said to be perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique.
Moreover, even if it initially was maximal, enforcing chordality may create a larger clique.
An undirected graph is perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique.
A maximal clique, sometimes called inclusion-maximal, is a clique that is not included in a larger clique.
