Georgia Institute of Technology College of Sciences

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham;  The Georgia Institute of Technology;  Emory University;  The University of Tennessee Knoxville.  The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology. Jason Cantarella (University of Georgia);   Meredith Casey (The Georga Institute of Technology);  Kirk Lancaster (Wichita State University); Junfang Li ( University of Alabama at Birmingham)  Jason Parsley (Wake Forest University);

Thomassen proved that there are only finitely many 6-critical graphs embeddable on a fixed surface. He also showed that planar graphs are 5-list-colorable. We develop new techniques to prove a general theorem for 5-list-coloring graphs embedded on a fixed surface. Namely, for every surface S and every integer C > 0, there exists D such that the following holds: Let G be a graph embedded in a surface S with edge-width at least D and a list assignment L such that, for every vertex v in G, L(v) has size at least five. If F is a collection of any number of facial cycles of length at most C such that every two cycles in F are distance at least D apart and every cycle in F has a locally planar neighborhood up to distance D/2, then any proper L-coloring of F extends to an L-coloring of G. This theorem implies the following results. In what follows, let S be a fixed surface, G be a graph embedded in S (except in 4, where G is drawn in S) and L a list assignment such that, for every vertex v of G, L(v) has size at least five. 1. If G has large edge-width, then G is 5-list-colorable. (Devos, Kawarabayashi and Mohar) 2. There exists only finitely many 6-list-critical graphs embeddable in S. (Conjectured by Thomassen, Proof announced by Kawarabayashi and Mohar) As a corollary, there exists a linear-time algorithm for deciding 5-list-colorability of graphs embeddable on S. Furthermore, we exhibit an explicit bound on the size of such graphs. 3. There exists D(S) such that the following holds: If X is a subset of the vertices of G that are pairwise distance at least D(S) apart, then any L-coloring of X extends to an L-coloring G. For planar graphs, this was conjectured by Albertson and recently proved by Dvorak, Lidicky, Mohar, and Postle. 4. There exists D(S) such that the following holds: If G is a graph drawn in S with face-width at least D(S) such that any pair of crossings is distance at least D apart, then G is L-colorable. For planar graphs, this was recently proved by Dvorak, Lidicky and Mohar. Joint work with Robin Thomas.

We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value proved by Erdos et al in 1989 is 22. It is conjectured that three vertices suffice. This is joint work with Daniel Kral, Chun-Hung Liu, Jean-Sebastien Sereni, and Zelealem Yilma.