# chromatic number

The chromatic number of a graph is the minimum number of colours required to colour it.

Consider the following graph:

 $\xymatrix{{\color[rgb]{1,0,0}A}\ar@{-}[r]&{\color[rgb]{0,0,1}B}\ar@{-}[r]\ar@{% -}[d]&{\color[rgb]{1,0,0}C}\ar@{-}[r]\ar@{-}[dl]&{\color[rgb]{0,1,0}F}\ar@{-}[% dl]\\ &{\color[rgb]{0,1,0}D}\ar@{-}[r]&{\color[rgb]{0,0,1}E}&}$

This graph has been coloured using $3$ colours. Furthermore, it’s clear that it cannot be coloured with fewer than $3$ colours, as well: it contains a subgraph ($BCD$) that is isomorphic to the complete graph of $3$ vertices. As a result, the chromatic number of this graph is indeed $3$.

This example was easy to solve by inspection. In general, however, finding the chromatic number of a large graph (and, similarly, an optimal colouring) is a very difficult (NP-hard) problem.

Title chromatic number ChromaticNumber 2013-05-16 21:23:44 2013-05-16 21:23:44 mathcam (2727) unlord (1) 9 mathcam (1) Definition msc 05C15 ChromaticNumberAndGirth FourColorConjecture ChromaticPolynomial ChromaticNumberOfASpace