Supereulerian 2-edge-coloured graphs

Abstract

A 2-edge-coloured graph G is supereulerian if G contains a spanning closed trail in which the edges alternate in colours. An eulerian factor of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs which cover all the vertices of G such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test if a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2edge-coloured graph is (trail-)colour-connected if it contains a pair of alternating (u, v)-paths ((u, v)-trails) whose union is an alternating closed walk for every pair of distinct vertices u, v. A 2-edge-coloured graph is M-closed if xz is an edge of G whenever some vertex u is joined to both x and z by edges of the same colour. M-closed 2-edge-coloured graphs, introduced in [11], form a rich generalization of 2-edge-coloured complete graphs. We show that if G is an extension of an M-closed 2-edge-coloured complete graph, then G is supereulerian if and only if G is trail-colourconnected and has an eulerian factor. We also show that for general 2-edge-coloured graphs it is NP-complete to decide whether the graph is supereulerian. Finally we pose a number of open problems. Keywords: 2-edge-coloured graph; alternating hamiltonian cycle; supereulerian; alternating cycle; eulerian factor; extension of a 2-edge-coloured graph.

The Complexity of Routing Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

“…The problem of properly colored Hamiltonian cycles was the first problem explicitly studied on edge-colored graphs and this problem and its variants (such as longest elementary cycle or spanning trails among many others) are especially well studied in the literature. We refer the reader to [30] or [5] for surveys on these problems and to [4], [13], [14], [31], [40] or [41] for recent developments.…”