Properly Coloured Hamiltonian Paths in Edge-coloured Complete Graphs

13 years 4 months ago

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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.

Jørgen Bang-Jensen, Gregory Gutin, Anders Y

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