Eulerian and Bipartite Orientable Matroids

Further work of Brylawski and Heron (see [4, p. 315]) explores other characterizations of Eulerian binary matroids. They showed, independently, that a binary matroid M is Eulerian if and only if its dual, M∗, is a binary affine matroid. More recently, Shikare and Raghunathan [5] have shown that a binary matroid M is Eulerian if and only if the number of independent sets of M is odd. This chapter is concerned with extending characterizations of Eulerian graphs via orientations. An Eulerian tour of a graph G induces an orientation with the property that every cocircuit (minimal edge cut) in G is traversed an equal number of times in each direction. In this sense, we can say that the orientation is balanced. Applying duality to planar graphs, these notions produce characterizations of bipartite graphs. Indeed the notions of flows and colourings of regular matroids can be formulated in terms of orientations, as was observed by Goddyn et al. [2]. The equivalent connection for graphs had been made by Minty [3]. In this chapter, we further extend these notions to oriented matroids. Informally, an oriented matroid is a matroid together with additional sign information. This is roughly analogous to orienting the edges in an undirected graph. We assume that the reader is familiar with basic matroid theory. In Section 2.1, we develop a view of oriented matroids which is suited to our purposes, and which should be accessible to a reader familiar with graphs and matroids at the graduate level.