chromatic number

plural chromatic numbers

1. graph theoryThe smallest number of colours needed to colour a given graph (i.e., to assign a colour to each vertex such that no two vertices connected by an edge have the same colour).

   The chromatic number of a complete graph K#95;n is n; the chromatic number of a bipartite graph K#95;#123;n,m#125; is 2.

   3 more examples

   The chromatic number of a graph G is the minimum value of k for which G is k-colourable, and is denoted by #92;chi(G).[…]A more essential use of the chromatic number was made by Gallai (1968a) and Roy (1967), who discovered a simple relationship between the chromatic number of a digraph and the length of a longest directed path in the digraph, where the chromatic number #92;chi(D) of a digraph D is defined to be the chromatic number of its underlying graph G(D).

   1995, J. A. Bondy, “1: Basic Graph Theory: Paths and Circuits”, in Ronald L. Graham, Martin Grötschel, László Lovász, editors, Handbook of Combinatorics, Volume 1, Elsevier (North-Holland), page 48

   2004, Monia Discepoli, Ivan Gerace, Riccardo Mariani, Andrea Remigi, A Spectral Technique to Solve the Chromatic Number Problem in Circulant Graphs, Antonio Laganà, et al. (editors), Computational Science and Its Applications, ICCSA 2004: International Conference, Proceedings, Part 3, Springer, LNCS 3045, page 745, The CHROMATIC NUMBER is the minimum number of colors by means of which it is possible to color a graph in such a way that each vertex has a different color with respect to the adjacent vertices. Such a problem is an NP-hard problem [14] and [it] is even hard to obtain a good approximation of the solution in a polynomial time [17]. Although in a lot of computational problems the cost decreases when these problems are restricted to circulant graphs [6, 9], the CHROMATIC NUMBER problem is NP-hard even restrecting to circulant graphs [9]. Moreover the problem of finding a good approximation of the CHROMATIC NUMBER problem on circulant graphs is also NP-hard.

   2009, Gary Chartrand, Ping Zhang, Chromatic Graph Theory, Taylor & Francis Group (CRC Press / Chapman & Hall), page 149, There is no general formula for the chromatic number of a graph. Consequently, we will often be concerned and must be content with (1) determining the chromatic number of some classes of interest and (2) determining upper and/or lower bounds for the chromatic number of a graph.

   Synonyms: vertex chromatic number