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Series: Graph Theory Seminar

Steinberg's Conjecture states that any planar graph without cycles of
length four or five is three colorable. Borodin, Glebov, Montassier,
and Raspaud showed that planar graphs without cycles of length four,
five, or seven are three colorable and Borodin and Glebov showed that
planar graphs without five cycles or triangles at distance at most two
apart are three colorable. We prove a statement similar to both of
these results: that any planar graph with no cycles of length four
through six or cycles of length seven with incident triangles distance
exactly two apart are three colorable. Special thanks to Robin Thomas
for substantial contributions in the development of the proof.

Series: Graph Theory Seminar

A characterization of graphs without an odd cycle is easy, of course,it is exactly bipartite. However, graphs without two vertex disjoint oddcycles are not so simple. Lovasz is the first to give a proof of the twodisjoint odd cycles theorem which characterizes internally 4-connectedgraphs without two vertex disjoint odd cycles. Note that a graph $G$ iscalled internally 4-connected if $G$ is 3-connected, and all 3-cutseparates only one vertex from the other.However, his proof heavily depends on the seminal result by Seymour fordecomposing regular matroids. In this talk, we give a new proof to thetheorem which only depends on the two paths theorem, which characterizesgraphs without two disjoint paths with specified ends (i.e., 2-linkedgraphs). In addition, our proof is simpler and shorter.This is a joint work with K. Kawarabayashi (National Institute ofInformatics).

Series: Graph Theory Seminar

The property that a graph has an embedding in projective plane is closed
under taking minors. So by the well known theorem of Robertson and Seymour,
there exists a finite list of minor-minimal graphs, call it L, such that a
given graph G is projective planar if and only if G does not contain any
graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found
35 graphs in L, and Archdeacon proved that those are all the members of L.
In this talk we show a new
strategy for finding the list L. Our approach is based on conditioning on
the connectivity of a member of L. Assume G is a member of L. If G is not
3-connected then the structure of G is well understood. In the case that G
is 3-connected, the problem breaks down into two main cases, either G has an
internal separation of order three or G is internally 4-connected . In this
talk we find the set of all 3-connected minor minimal non-projective planar
graphs with an internal 3-separation.
This is joint work with Luke Postle and Robin Thomas.

Series: Graph Theory Seminar

This lecture will conclude the series. In a climactic finish the speaker will prove the Unique Linkage Theorem, thereby completing the proof of correctness of the Disjoint Paths Algorithm.

Series: Graph Theory Seminar

First-Fit is an online algorithm that partitions the elements of a poset
into chains. When presented with a new element x, First-Fit adds x
to the first chain whose elements are all comparable to x. In 2004,
Pemmaraju, Raman, and Varadarajan introduced the Column Construction
Method to prove that when P is an interval order of width w,
First-Fit partitions P into at most 10w chains. This bound was
subsequently improved to 8w by Brightwell, Kierstead, and Trotter, and
independently by Narayanaswamy and Babu.
The poset r+s is the disjoint union of a chain of size r and a chain
of size s. A poset is an interval order if and only if it does not
contain 2+2 as an induced subposet. Bosek, Krawczyk, and Szczypka
proved that if P is an (r+r)-free poset of width w, then First-Fit
partitions P into at most 3rw^2 chains and asked whether the bound
can be improved from O(w^2) to O(w). We answer this question in
the affirmative. By generalizing the Column Construction Method, we
show that if P is an (r+s)-free poset of width w, then First-Fit
partitions P into at most 8(r-1)(s-1)w chains.
This is joint work with Gwena\"el Joret.

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

Series: Graph Theory Seminar

Series: Graph Theory Seminar

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.