# Tight Bounds on The Clique Chromatic Number

@article{Joret2021TightBO, title={Tight Bounds on The Clique Chromatic Number}, author={Gwena{\"e}l Joret and Piotr Micek and Bruce A. Reed and Michiel H. M. Smid}, journal={ArXiv}, year={2021}, volume={abs/2006.11353} }

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.

## 2 Citations

### Tight asymptotics of clique-chromatic numbers of dense random graphs

- Mathematics
- 2020

The clique chromatic number of a graph is the minimum number of colors required to assign to its vertex set so that no inclusion maximal clique is monochromatic. McDiarmid, Mitsche and Pra lat proved…

### The jump of the clique chromatic number of random graphs

- MathematicsArXiv
- 2021

This work resolves the nature of this polynomial ‘jump’ of the clique chromatic number of the random graph Gn,p around edge-probability p ≈ n−1/2 and goes beyond Janson’s inequality used in previous work and determines theClique chromatics number of Gn, p up to logarithmic factors for any edge- Probabilityp.

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