## Seminars and Colloquia by Series

### Induced subgraphs in graphs without linear and polynomial-size anticomplete sets

Series
Graph Theory Seminar
Time
Thursday, February 8, 2018 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sophie SpirklPrinceton University
The celebrated Erdos-Hajnal conjecture states that for every graph H, there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of size at least n^c, where n = |V(G)|. One approach for proving this conjecture is to prove that in every H-free graph G, there are two linear-size sets A and B such that either there are no edges between A and B, or every vertex in A is adjacent to every vertex in B. As is turns out, this is not true unless both H and its complement are trees. In the case when G contains neither H nor its complement as an induced subgraph, the conclusion above was conjectured to be true for all trees (Liebenau & Pilipczuk), and I will discuss a proof of this for a class of tree called "caterpillars". I will also talk about results and open questions for some variants, including allowing one or both of A and B to have size n^c instead of linear size, and requiring the bipartite graph between A and B to have high or low density instead of being complete or empty. In particular, our results improve the bound on the size of the largest clique or stable that must be present in a graph with no induced five-cycle. Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott, and Paul Seymour.

### Two-three linked graphs

Series
Graph Theory Seminar
Time
Thursday, November 30, 2017 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shijie XieMath, Gt
Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will complete a sketch of our arguments for characterizing when (G, a0, a1, a2, b1, b2) is feasible. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

### Two-three linked graphs

Series
Graph Theory Seminar
Time
Thursday, November 9, 2017 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shijie XieMath, GT
Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will prove the existence of 5-edge configurations in (G, a0, a1, a2, b1, b2). Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

### Two-three linked graphs

Series
Graph Theory Seminar
Time
Thursday, November 2, 2017 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shijie XieMath, GT
Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will introduce ideal frames, slim connectors and fat connectors. We will first deal with the ideal frames without fat connectors, by studying 3-edge and 5-edge configurations. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

### Two-three linked graphs

Series
Graph Theory Seminar
Time
Thursday, October 5, 2017 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shijie XieMath, GT
Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will describe the structure of G when (G, a0, a1, a2, b1, b2) is infeasible, using frames and connectors. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

### Stability results in graphs of given circumference

Series
Graph Theory Seminar
Time
Thursday, September 28, 2017 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jie MaUniversity of Science and Technology of China

### Elusive problems in extremal graph theory

Series
Graph Theory Seminar
Time
Thursday, May 18, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel KralUniversity of Warwick
We study the uniqueness of optimal configurations in extremal combinatorics. An empirical experience suggests that optimal solutions to extremal graph theory problems can be made asymptotically unique by introducing additional constraints. Lovasz conjectured that this phenomenon is true in general: every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints such that the resulting set is satisfied by an asymptotically unique graph. We will present a counterexample to this conjecture and discuss related results. The talk is based on joint work with Andrzej Grzesik and Laszlo Miklos Lovasz.

### The extremal function, Colin de Verdiere parameter, and chromatic number of graphs

Series
Graph Theory Seminar
Time
Thursday, March 16, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rose McCartyMath, GT
For a graph G, the Colin de Verdière graph parameter mu(G) is the maximum corank of any matrix in a certain family of generalized adjacency matrices of G. Given a non-negative integer t, the family of graphs with mu(G) <= t is minor-closed and therefore has some nice properties. For example, a graph G is planar if and only if mu(G) <= 3. Colin de Verdière conjectured that the chromatic number chi(G) of a graph satisfies chi(G) <= mu(G)+1. For graphs with mu(G) <= 3 this is the Four Color Theorem. We conjecture that if G has at least t vertices and mu(G) <= t, then |E(G)| <= t|V(G)| - (t+1 choose 2). For planar graphs this says |E(G)| <= 3|V(G)|-6. If this conjecture is true, then chi(G) <= 2mu(G). We prove the conjectured edge upper bound for certain classes of graphs: graphs with mu(G) small, graphs with mu(G) close to |V(G)|, chordal graphs, and the complements of chordal graphs.