Graph products and monochromatic multiplicities

Abstract

Arcane two-edge-colourings of complete graphs were described in [13], in which there are significantly fewer monochromaticK r 's than in a random colouring (so disproving a conjecture of Erdős [2]). Jagger, Šťovíček and Thomason [7] showed that the same colourings have fewer monochromaticG's than do random colourings for any graphG containingK 4.

The purpose of this note is to point out that these colourings are not as obscure as might appear. There is in fact a large, natural and easily described class of colourings of the above kind; the specific examples used in [13] and [7] fall into this class.

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Thomason, A. Graph products and monochromatic multiplicities. Combinatorica 17, 125–134 (1997). https://doi.org/10.1007/BF01196136

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Mathematics Subject Classification (1991)

  • 05 C 35
  • 05 C 55