Note edge-colourings of K n,n with no long two-coloured cycles

Abstract

Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Häggkvist in 1978 and, specifically, L(n, K n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K n,n with n colours having all two-coloured cycles of length ≤ 2q 2, for integers n in a set of density 1 − 3/(q − 1). One consequence is that L(n, K n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n − 4.

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Correspondence to Peter Dukes.

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Research of the first author is supported by NSERC.

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Dukes, P., Ling, A.C.H. Note edge-colourings of K n,n with no long two-coloured cycles. Combinatorica 28, 373–378 (2008). https://doi.org/10.1007/s00493-008-2272-6

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Mathematics Subject Classification (2000)

  • 05C15
  • 05B25