chromatic number

== English == === Noun === chromatic number (plural chromatic numbers) (graph theory) The 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). 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[sic] 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. ==== Usage notes ==== Not to be confused with chromatic index (aka edge-chromatic number), which is the equivalent minimum number for an edge colouring. The chromatic number of a graph G {\displaystyle G} is often denoted χ ( G ) {\displaystyle \chi (G)} . ==== Synonyms ==== (smallest number of colours needed to colour the vertices of a graph): vertex chromatic number (used to differentiate from edge chromatic number, etc.) ==== Derived terms ==== chromatic number problem edge chromatic number harmonious chromatic number total chromatic number ==== Related terms ==== achromatic number chromatic index chromatic polynomial ==== Translations ==== === See also === k-colouring k-chromatic === Further reading === Graph coloring on Wikipedia.Wikipedia Edge coloring on Wikipedia.Wikipedia Total coloring on Wikipedia.Wikipedia Hadwiger conjecture (graph theory) on Wikipedia.Wikipedia