A Note on Alternating Cycles in Edge-Coloured Graphs

Grossman and Häggkvist gave a sufficient condition under which a two-edge-coloured graph must have an alternating cycle (i.e., a cycle in which no two consecutive edges have the same colour). We extend their result to edge-coloured graphs with any number of colours. That is, we show that if there is no alternating cycle in an edge-coloured graphG, thenGcontains a vertexzsuch that no connected component ofG−zis joined tozwith edges of more than one colour. Our result implies that there is a polynomial-time algorithm for deciding whether an edge-coloured graph contains an alternating cycle.