On cycle-double covers of graphs of small oddness

Abstract The oddness of a multigraph was introduced by M. Kochol and the author. It is always a nonnegative even number and for cubic multigraphs G, it measures how far G is away from being 3-edge colourable. M. Kochol and the author proved that each bridgeless multigraph of oddness at most 2 contains a 5-CDC, i.e. a cycle-double cover consisting of at most 5 Eulerian subgraphs. In this paper, we extend this result to all bridgeless multigraphs of oddness at most 4 by providing some technics which perhaps will lead to even more extensions. As a consequence, each bridgeless multigraph containing a spanning tree with at most 3 endvertices has 5-CDC.