A Note on Alternating Cycles in Edge-Coloured Graphs

Grossman and HH aggkvist gave a characterisation of two-edge-coloured graphs, which have an alternating cycle (i.e. a cycle in which no two consecutive edges have the same colour). We extend their characterisa-tion to edge-coloured graphs, with any number of colours. That is we show that if there is no alternating cycle in an edge-coloured graph, G, then it contains a vertex z, such that there is no connected component of G ? z, which is joined to z with edges of diierent colors. Our extension implies a polynomial algorithm, for deciding if an edge-coloured graph contains an alternating cycle (see also 1]).