Multiplicities of subgraphs

Abstract

A former conjecture of Burr and Rosta [1], extending a conjecture of Erdős [2], asserted that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graphG which are monochromatic is at least the proportion found in a random colouring. It is now known that the conjecture fails for some graphsG, includingG=K p forp≥4.

We investigate for which graphsG the conjecture holds. Our main result is that the conjecture fails ifG containsK 4 as a subgraph, and in particular it fails for almost all graphs.

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Jagger, C., Šťovíček, P. & Thomason, A. Multiplicities of subgraphs. Combinatorica 16, 123–141 (1996). https://doi.org/10.1007/BF01300130

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Mathematics Subject Classification (1991)

  • 05 C 55
  • 05 C 35