Properly coloured Hamiltonian cycles in edge-coloured complete graphs

Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{o}\mathrm{n}}(K_n^c)$ denote the largest number of edges of the same colour incident with a vertex of $K_n^c$. A properly coloured cycle is a cycle such that no two adjacent edges have the same colour. In 1976, Bollob'as and Erd\H{o}s conjectured that every $K_n^c$ with $\Delta_{\rm mon}(K_n^c) < \lfloor n/2 \rfloor$ contains a properly coloured Hamiltonian cycle. In this paper, we show that for any $\epsilon >0$, there exists an integer $n_0$ such that every $K_n^c$ with ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{o}\mathrm{n}}(K_n^c)<(1\mathrm{/}2-\epsilon )n$ and $n\ge n_0$ contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin. Hence, the conjecture of Bollob'as and Erd\H{o}s is true asymptotically.