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 ( $B C D$ ) 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.