Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 3
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Ma, Jie and Naves, Humberto 2015. Maximizing proper colorings on graphs. Journal of Combinatorial Theory, Series B, Vol. 115, p. 236.

    Beaudin, Laura Ellis-Monaghan, Joanna Pangborn, Greta and Shrock, Robert 2010. A little statistical mechanics for the graph theorist. Discrete Mathematics, Vol. 310, Issue. 13-14, p. 2037.

    Sakaloglu, A. and Satyanarayana, A. 1995. Graphs with least number of colorings. Journal of Graph Theory, Vol. 19, Issue. 4, p. 523.

  • Proceedings of the Edinburgh Mathematical Society, Volume 2, Issue 2
  • June 1930, pp. 83-91

On the Number of Ways of Colouring a Map


It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the following

Theorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then


This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

C. N. Reynolds , On the Problem of Colouring Maps in Four Colours, II, Annals of Mathematics, vol. 28, second series, 477492.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *