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Few-colour Hamilton cycle in optimally edge-coloured complete graphs

Determine whether every optimally edge-coloured complete graph Kn contains a Hamilton cycle using only O(log n) colours.

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Background

In an optimally edge-coloured Kn (using χ′(Kn) colours), Akbari et al. proved a bound of O(√n) colours for a Hamilton cycle and conjectured the optimal logarithmic bound.

The paper’s Theorem 1.5 on spectral expanders yields Theorem 1.10, confirming the conjectured O(log n) bound through a pseudorandom reduction.

References

Akbari, Etesami, Mahini, and Mahmoody [3] proved that √ny optimally coloured n-vertex complete graph K han a Hamilton cycle containing edges of at most 8 n colours, and conjectured there should always be such a cycle using only O(logn) colours, which would be best possible up to a multiplicative constant.

— Hamiltonicity of expanders: optimal bounds and applications (2402.06603 - Draganić et al., 9 Feb 2024) in Section 1.1