Matroid and Tutte-connectivity in infinite graphs

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.Comment: 11 page

Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices

We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.Comment: 43 page

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

Axioms for infinite matroids

We give axiomatic foundations for non-finitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig