Some inequalities for the Tutte polynomial

We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.Comment: 17 page

Eulerian and bipartite orientable matroids

Welsh [6] extended to the class of binary matroids a well-known theorem regarding Eulerian graphs. Theorem 2.1 Let M be a binary matroid. The ground set E(M) can be partitioned into circuits if and only if every cocircuit of M has even cardinality. 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. 2.1 Orientations without vertices The concept of orienting a graph can be understood by a child. Since a matroid has no vertices, one must work harder to understand oriented matroids. There are two equivalent definitions of oriented matroids: axiomatic and geometric. Each view offers advantages in understanding and working with these objects