Eulerian colorings and the bipartizing matchings conjecture of Fleischner

A trigraph is a multigraph with half-edges colored in three colors. We introduce the notion of an Eulerian coloring of a trigraph and show that the existence of two orthogonal Eulerian colorings in a special class of trigraphs is closely related to the bipartizing matchings conjecture of Fleischner, and hence to the cycle double cover conjecture and Tutte’s 5-flow conjecture. We prove that every trigraph has an Eulerian coloring and that a rainbow cubic trigraph has a pair of orthogonal Eulerian colorings if and only if it has a perfect matching.