Properly coloured Hamiltonian cycles in edge-coloured complete graphs

Abstract

Let K c n be an edge-coloured complete graph on n vertices. Let Δmon(Kc n) denote the largest number of edges of the same colour incident with a vertex of Kc n. A properly coloured cycleis a cycle such that no two adjacent edges have the same colour. In 1976, BollobÁs and ErdŐs[6] conjectured that every Kc n with Δmon(Kc n)<⌊n/2⌋contains a properly coloured Hamiltonian cycle. In this paper, we show that for any ε>0, there exists an integer n0 such that every Kc n with Δmon(Kc n)<(1/2–ε)n and n≥n0 contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin [1]. Hence, the conjecture of BollobÁs and ErdŐs is true asymptotically.

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Correspondence to Allan Lo.

Additional information

The research leading to these results was supported by the European Research Council under the ERC Grant Agreement no. 258345.

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Lo, A. Properly coloured Hamiltonian cycles in edge-coloured complete graphs. Combinatorica 36, 471–492 (2016). https://doi.org/10.1007/s00493-015-3067-1

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C38