Eulerian colorings and the bipartizing matchings conjecture of Fleischner

Fleischner [3] has proposed a novel approach to several deep graph-theoretic conjectures, such as the Cycle Double Cover Conjecture [9,10] or the 5-Flow Conjecture of Tutte [12]. A key ingredient in this approach is the Bipartizing Matchings Conjecture (Conjecture 1.2 below, abbreviated BM Conjecture). In this paper, we introduce the notion of an Eulerian coloring of a trigraph, providing a setting that allows a generalization of the Bipartizing Matchings Conjecture and inspires new questions. We also prove the counterparts of major known facts about bipartizing matchings in the more general context.