Properly coloured Hamiltonian paths in edge-coloured complete graphs

We consider edge-coloured complete graphs. A path or cycle Q is called properly coloured (PC) if any two adjacent edges of Q differ in colour. Our note is inspired by the following conjecture by B. Bollobás and P. Erdős (1976): if G is an edge-coloured complete graph on n vertices in which the maximum monochromatic degree of every vertex is less than ⌞n2⌟, then G contains a PC Hamiltonian cycle. We prove that if an edge-coloured complete graph contains a PC 2-factor then it has a PC Hamiltonian path. R. Häggkvist (1996) announced that every edge-coloured complete graph satisfying Bollobás-Erdős condition contains a PC 2-factor. These two results imply that every edge-coloured complete graph satisfying Bollobás-Erdős condition has a PC Hamiltonian path.