Properly coloured Hamiltonian cycles in edge-coloured complete graphs

Abstract

Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let ${\Delta }_{\mathrm{mon}}(K_n^c)$ 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, Bollobás and Erdős conjectured that every $K_n^c$ with ${\Delta }_{\mathrm{mon}}(K_n^c)<\lfloor n/2\rfloor$ 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 ${\Delta }_{\mathrm{mon}}(K_n^c)<(1/2-\epsilon )n$ and $n\ge n_0$ contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin. Hence, the conjecture of Bollobás and Erdős is true asymptotically.

18 pages, minor revision. Now accepted for publication in Combinatorica

Cited by in corpus (1)

• An edge-coloured version of Dirac's theorem