Tight Bounds on The Clique Chromatic Number

  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},
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. 

Tight asymptotics of clique-chromatic numbers of dense random graphs

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

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.



Clique coloring of binomial random graphs

It is seen that the typical clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n) forms an intriguing step function.

The Grötzsch Theorem for the Hypergraph of Maximal Cliques

The Grotzsch Theorem is extended to list colorings by proving that the clique hypergraph of every planar graph is 3-colorable and 4-choosability of ${\cal H}(G)$ is established for the class of locally planar graphs on arbitrary surfaces.

On the divisibility of graphs

Fibres and ordered set coloring

The Ramsey Number R(3, t) Has Order of Magnitude t2/log t

It is proved that R(3, t) is bounded below by (1 – o(1))t/2/log t times a positive constant, and it follows that R (3), the Ramsey number for positive integers s and t, has asymptotic order of magnitude t2/ log t.

Two-colouring all two-element maximal antichains

ω-Perfect graphs

Abstractω-Perfect graph is defined and some classes of ω-perfect graphs are described, although the characterization of the complete class of ω-perfect graphs remains an open question. A bound on the

The list chromatic number of graphs with small clique number

and P