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Series: Graph Theory Seminar

The k-disjoint paths problem takes as input a graph G and k pairs
of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist
internally disjoint paths P_1,..., P_k such that the endpoints of P_i
are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete
when k is allowed to be part of the input, Robertson and Seymour showed
that there exists a polynomial time algorithm for fixed values of k. The
existence of such an algorithm is the major algorithmic result of the
Graph Minors series. The original proof of Robertson and Seymour relies
on the whole theory of graph minors, and consequently is both quite
technical and involved. Recent results have dramatically simplified the
proof to the point where it is now feasible to present the proof in its
entirety. This seminar series will do just that, with the level of detail
aimed at a graduate student level.

Series: Graph Theory Seminar

A graph is k-critical if it is not (k-1)-colorable but every proper subgraph is. In 1963, Gallai conjectured that every k-critical graph G of order n has at least (k-1)n/2 + (k-3)(n-k)/(2k-2) edges. The currently best known results were given by Krivelevich for k=4 and 5, and by Kostochka and Stiebitz for k>5. When k=4, Krivelevich's bound is 11n/7, and the bound in Gallai's conjecture is 5n/3 -2/3. Recently, Farzad and Molloy proved Gallai's conjecture for k=4 under the extra condition that the subgraph induced by veritces of degree three is connected. We will review the proof given by Krivelevich, and the proof given by Farzad and Molloy in the seminar.

Series: Graph Theory Seminar

Series: Graph Theory Seminar

This will be a continuation from last week. We extend the theory of infinite matroids recently developed by Bruhn et al to a well-known classical result in finite matroids while using the theory of connectivity for infinitematroids of Bruhn and Wollan. We prove that every infinite connected matroid M determines a graph-theoretic decomposition tree whose vertices correspond to minors of M that are3-connected, circuits, or cocircuits, and whose edges correspond to 2-separations of M. Tutte and many other authors proved such a decomposition for finite graphs; Cunningham andEdmonds proved this for finite matroids and showed that this decomposition is unique if circuits and cocircuits are also allowed. We do the same for infinite matroids. The knownproofs of these results, which use rank and induction arguments, do not extend to infinite matroids. Our proof avoids such arguments, thus giving a more first principles proof ofthe finite result. Furthermore, we overcome a number of complications arising from the infinite nature of the problem, ranging from the very existence of 2-sums to proving the treeis actually graph-theoretic.

Series: Graph Theory Seminar

Series: Graph Theory Seminar

We discuss open research questions surrounding the traveling salesman problem.
A focus will be on topics having potential impact on the computational
solution of large-scale problem instances.

Series: Graph Theory Seminar

Tree-width is a well-known metric on undirected graphs that measures how tree-like a
graph is and gives a notion of graph decomposition that proves useful in
fixed-parameter tractable (FPT) algorithm development. In the directed setting, many
similar notions have been proposed - none of which has been accepted widely as a
natural generalization of tree-width. Among the many suggested equivalent parameters
were the "directed tree-width" by Johnson et al, and DAG-width by Berwanger et al and
Odbrzalek.
In this talk, I will present a recent paper by Hunter and Kreutzer, that defines
another such directed width parameter, celled "kelly-width". I will discuss the
equivalent complexity measures for graphs such as elimination orderings, k-trees and
cops and robber games and study their natural generalizations to digraphs. I will
discuss its usefulness by discussing potential applications including polynomial-time
algorithms for NP-complete problems on graphs of bounded Kelly-width (FPT). I will
also briefly discuss our work in progress (joint with Shiva Kintali) towards
designing an approximation algorithm for Kelly Width.

Series: Graph Theory Seminar

Rota asked in the 1960's how one might construct an axiom system for
infinite matroids. Among the many suggested answers
were the B-matroids of Higgs. In 1978, Oxley proved that any infinite
matroid system with the notions of duality and minors must be equivalent to
B-matroids. He also provided a simpler mixed basis-independence axiom system
for B-matroids, as opposed to the complicated closure system developed by
Higgs. In this talk, we examine a recent paper of Bruhn et al that gives
independence, basis, circuit, rank, and closure axiom systems for
B-matroids. We will also discuss some open problems for infinite matroids.

Series: Graph Theory Seminar

A graph is k-choosable if it can be properly colored from any assignment of lists of colors of length at least k to its vertices. A well-known results of Thomassen state that every planar graph is 5-choosable and every planar graph of girth 5 is 3-choosable. These results are tight, as shown by constructions of Voigt. We review some new results in this area, concerning 3-choosability of planar graphs with constraints on triangles and 4-cycles.

Series: Graph Theory Seminar

We present some geometric properties of the Laplacian lattice and the lattice of integer flows of a given graph and discuss some applications and open problems.