Alternating cycles and trails in 2-edge-coloured complete multigraphs

We consider edge-coloured multigraphs. A trail in such a multigraph is alternating if its successive edges differ in colour. Let G be a 2-edge-coloured complete graph and let M be a 2-edge-coloured complete multigraph. M. Bankfalvi and Zs. Bankfalvi [2] obtained a necessary and sufficient condition for G to have a Hamiltonian alternating cycle. Generalizing this theorem, P. Das and S.B. Rao [7] characterized those G which contain a closed alternating trail visiting each vertex v in G exactly f(v) > 0 times. We solve the more general problem of determining the length of a longest closed alternating trail Tf visiting each vertex v in M at most f(v) > 0 times. Our result is a generalization of a theorem by R. Saad [18] that determines the length of a longest alternating cycle in G. We prove the existence of a polynomial algorithm for finding the desired trail Tf . In particular, this provides a solution to a question in [18].

• Type: Journal
• Year: 1998
• Where: DM
• Authors: Jørgen Bang-Jensen, Gregory Gutin
• Keywords: 2-edge-coloured Complete Graph, DM 1998, Hamiltonian Alternating Cycle, Trail