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. Bollob´as and P. Erd˝os (1976) : if G is an edge-coloured complete graph on n vertices in which the maximum monochromatic degree of every vertex is less than n/2 , 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. H¨aggkvist (1996) announced that every edge-coloured complete graph satisfying Bollob´as-Erd˝os condition contains a PC 2-factor. These two results imply that every edge-coloured complete graph satisfying Bollob´as-Erd˝os condition has a PC Hamiltonian path.

• Type: Journal
• Year: 1998
• Where: DAM
• Authors: Jørgen Bang-Jensen, Gregory Gutin, Anders Yeo
• Keywords: Complete Graph, DAM 1998, Edge-coloured Complete Graph, Maximum Monochromatic Degree