Tuesday, April 12, 3pm ET (8pm BST, 7am Wed NZST)
Carolyn Chun, United States Naval Academy
Lattice path matroids, lattice path polymatroids, and excluded minors

Abstract:

We define lattice path matroids, polymatroids, Boolean polymatroids, and lattice path polymatroids, which are a subclass of Boolean polymatroids.  We give an excluded minor characterization for lattice path polymatroids, based on a proof where the main tool was Venn diagrams!  There are infinitely many excluded minors for lattice path polymatroids, but they fall into a small number of easily-described types.

Tuesday, April 5, 3pm ET (8pm BST, *7am* Wed NZST)
Lise Turner, University of Waterloo
A local version of Hadwiger’s Conjecture

Abstract:

In 1943, Hadwiger famously conjectured that graphs with no $K_t$ minors are $t-1$ colourable. There has also been significant interest in several variants of the problem, such as list colouring or only forbidding certain classes of minors. We propose a local version where all balls of radius $O(\log v(G))$ must be $K_t$-minor free but the graph as a whole may not be. We prove list colouring results for these graphs equivalent to the best known results for $K_t$-minor free graphs for $t\leq 5$ and large $t$. In the process, we provide some efficient distributed algorithms for finding such colourings.

Joint work with Benjamin Moore and Luke Postle.

Tuesday, March 29, 3pm ET (8pm GMT, 8am Wed NZDT)
Zishen Qu, University of Waterloo
Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs

Abstract:

Finding sufficient conditions for a class of graphs to be Hamiltonian is an old problem, with a wide variety of conditions such as Dirac’s degree condition and Whitney’s theorem on 4-connected planar triangulations. We discuss some past results on sufficient conditions for Hamiltonicity involving the exclusion of fixed induced subgraphs, and some properties of the graphs involved in such results. In 1981 Duffus, Gould, and Jacobson showed that any connected graph that does not contain a claw or a net as an induced subgraph has a Hamiltonian path. We aim to find an analogous result for Hamiltonian cycles. In particular, we would like to find a set of graphs $S$ which are 2-connected, non-Hamiltonian, and every proper 2-connected induced subgraph is Hamiltonian such that every 2-connected $S$-free graph is Hamiltonian. In joint work with Joseph Cheriyan, Sepehr Hajebi, and Sophie Spirkl, we show that the classes of 2-connected split graphs and 2-connected triangle-free graphs can be characterised in this fashion.