Tight Bounds on the Clique Chromatic Number

TL;DR

This paper establishes tight bounds on the clique chromatic number of graphs, showing it scales with maximum degree and number of vertices in a way that is asymptotically optimal.

Contribution

The authors prove tight bounds on the clique chromatic number, linking it to maximum degree and number of vertices, advancing understanding of graph coloring complexities.

Findings

Clique chromatic number is O(Δ / log Δ) for graphs with maximum degree Δ.

Clique chromatic number is O(√(n / log n)) for n-vertex graphs.

Both bounds are proven to be tight.

Abstract

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.