, Volume 5, Issue 2, pp 121–126

The chromatic number of the product of two 4-chromatic graphs is 4

  • Mohamed El-Zahar
  • Norbert Sauer

DOI: 10.1007/BF02579374

Cite this article as:
El-Zahar, M. & Sauer, N. Combinatorica (1985) 5: 121. doi:10.1007/BF02579374


For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a)g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs.

AMS subject classification (1980)

05 C 15

Copyright information

© Akadémiai Kiadó 1985

Authors and Affiliations

  • Mohamed El-Zahar
    • 1
  • Norbert Sauer
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada