Excluded Structures for Pinch-graphic Matroids and Tournaments

Abstract

In this thesis, we study several problems related to excluded structures in matroids and tournaments. For matroids, we focus on finding the excluded minors for pinch-graphic matroids. For tournaments, we bound the size of excluded induced subgraphs of k-degenerate tournaments and we bound the number of backedge graphs of a tournament that are P3-free or Kt-free. Furthermore, we introduce a new way to color the arcs of a tournament and we propose a conjecture that connects this coloring to tournaments that do not contain Paley tournaments as an induced subgraph.

For pinch-graphic matroids, we prove that if M is minimally non-pinch-graphic and M is not 3-connected, then |M| ≤ 20. Furthermore, we characterize all minimally non-pinch-graphic matroids that are not 3-connected. For the 3-connected case, we prove that in a special case, a 3-connected but not internally 4-connected minimally non-pinch-graphic matroid has at most 21 elements. We also improve a characterization of 3-separations in pinch-graphic matroids by Guenin and Heo.

For tournaments, we show that an excluded induced subgraph for 1-in(out)-degenerate tournaments has at most (k² + 5k + 6) / 2 vertices, and an excluded induced subgraph for 1-degenerate tournaments has at most k² + 5k + 6 vertices. We also prove that a tournament with n vertices can have at most exponentially many P3-free or Kt-free backedge graphs, despite there being n! backedge graphs for a tournament with n vertices. We then introduce a new way to arc-color a tournament T with the corresponding chromatic number denoted as χ_Δ(T). The rule is to arc-color a tournament such that every directed cycle of length 3 gets 3 distinct colors. We prove that for tournaments T with n vertices, the growth rate of χ_Δ(T) is Θ(log n).