Tight Bounds on the Clique Chromatic Number
Copyright policy )Tight Bounds on the Clique Chromatic Number
The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.
FOS: Computer and information sciences, clique colouring, Discrete Mathematics (cs.DM), Géométrie, Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.), Coloring of graphs and hypergraphs, Informatique mathématique, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), perfect graphs, Computer Science - Discrete Mathematics
FOS: Computer and information sciences, clique colouring, Discrete Mathematics (cs.DM), Géométrie, Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.), Coloring of graphs and hypergraphs, Informatique mathématique, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), perfect graphs, Computer Science - Discrete Mathematics
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