chromatic number

English

A minimal colouring of a graph whose chromatic number is 3

Noun

chromatic number (plural chromatic numbers)

  1. (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).
    The chromatic number of a complete graph is ; the chromatic number of a bipartite graph is 2.
    • 1995, J. A. Bondy, 1: Basic Graph Theory: Paths and Circuits, Ronald L. Graham, Martin Grötschel, László Lovász (editors), Handbook of Combinatorics, Volume 1, Elsevier (North-Holland), page 48,
      The chromatic number of a graph is the minimum value of for which is -colourable, and is denoted by . [] 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 of a digraph is defined to be the chromatic number of its underlying graph .
    • 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 is often denoted .

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
  • achromatic number
  • chromatic index
  • chromatic polynomial

Translations

See also

  • k-colouring
  • k-chromatic

Further reading

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