A note on approximating theb-chromatic number

On approximating the b-chromatic number

We consider the problem of approximating the b-chromatic number of a graph. We show that there is no constant ε > 0 for which this problem can be approximated within a factor of 120/113− ε in polynomial time, unless P = NP. This is the first hardness result for approximating the b-chromatic number.

A note on cyclic chromatic number

A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χc(G) ≤ ∆∗ + 2 for any 3-connected plane graph G with maximum face degree ∆∗. It is known that th...

Improved Hardness of Approximating Chromatic Number

We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2 1/3 colors. This improves the previous result of K versus K in Khot [14].

On the Hardness of Approximating the Chromatic Number

A note on generalized chromatic number and generalized girth

Erdős proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers. Generalized graph coloring describes the partitioning of the vertices into classes whose induced subgraphs satisfy particular constraints. When P is a family of graphs, the P chromatic number of a graph G, written χP, is the minimum size of a pa...