## Seminars and Colloquia by Series

Thursday, April 19, 2018 - 13:30 , Location: Skiles 005 , Alexander Hoyer , Math, GT , Organizer: Robin Thomas
Györi and Lovasz independently proved that a k-connected graph can be partitioned into k subgraphs, with each subgraph connected, containing a prescribed vertex, and with a prescribed vertex count. Lovasz used topological methods, while Györi found a purely graph theoretical approach. Chen et al. later generalized the topological proof to graphs with weighted vertices, where the subgraphs have prescribed weight sum rather than vertex count. The weighted result was recently proven using Györi's approach by Chandran et al. We will use the Györi approach to generalize the weighted result slightly further. Joint work with Robin Thomas.
Thursday, April 12, 2018 - 13:30 , Location: Skiles 005 , Youngho Yoo , Math, GT , Organizer: Robin Thomas
A classic theorem of Mader gives the extremal functions for graphs that do not contain the complete graph on p vertices as a minor for p up to 7. Motivated by the study of linklessly embeddable graphs, we present some results on the extremal functions of apex graphs with respect to the number of triangles, and on triangle-free graphs with excluded minors. Joint work with Robin Thomas.
Thursday, March 8, 2018 - 13:30 , Location: Skiles 005 , Alexander Hoyer , Math, GT , Organizer: Robin Thomas
For a graph G, a set of subtrees of G are edge-independent with root r ∈ V(G) if, for every vertex v ∈ V(G), the paths between v and r in each tree are edge-disjoint. A set of k such trees represent a set of redundant broadcasts from r which can withstand k-1 edge failures. It is easy to see that k-edge-connectivity is a necessary condition for the existence of a set of k edge-independent spanning trees for all possible roots. Itai and Rodeh have conjectured that this condition is also sufficient. This had previously been proven for k=2, 3. We prove the case k=4 using a decomposition of the graph similar to an ear decomposition. Joint work with Robin Thomas.
Thursday, March 1, 2018 - 13:30 , Location: Skiles 005 , Alexander Barvinok , University of Michigan , , Organizer: Prasad Tetali
This is Lecture 2 of a series of 3 lectures by the speaker. See the abstract on Tuesday's ACO colloquium of this week. (Please note that this lecture will be 80 minutes' long.)
Thursday, February 8, 2018 - 13:30 , Location: Skiles 005 , Sophie Spirkl , Princeton University , Organizer: Robin Thomas
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.
Thursday, November 30, 2017 - 13:30 , Location: Skiles 005 , Shijie Xie , Math, Gt , Organizer: Robin Thomas
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.
Thursday, November 9, 2017 - 13:30 , Location: Skiles 005 , Shijie Xie , Math, GT , Organizer: Robin Thomas
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.
Thursday, November 2, 2017 - 13:30 , Location: Skiles 005 , Shijie Xie , Math, GT , Organizer: Robin Thomas
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.
Thursday, October 5, 2017 - 13:30 , Location: Skiles 005 , Shijie Xie , Math, GT , Organizer: Robin Thomas
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.
Thursday, September 28, 2017 - 13:30 , Location: Skiles 005 , Jie Ma , University of Science and Technology of China , Organizer: Xingxing Yu
In this talk we will discuss some Tur\'an-type results on graphs with a given circumference. Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$ by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$ vertices of the clique, and let $f(n,k,c)=e(W_{n,k,c})$. Kopylov proved in 1977 that for $c\max\{f(n,3,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)\}$, then either $G$ is a subgraph of $W_{n,2,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, or $c$ is odd and $G$ is a subgraph of a member of two well-characterized families which we define as $\mathcal{X}_{n,c}$ and $\mathcal{Y}_{n,c}$. We extend and refine their result by showing that if $G$ is a 2-connected graph on $n$ vertices with minimum degree at least $k$ and circumference $c$ such that $10\leq c\max\{f(n,k+1,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)\}$, then one of the following holds:\\ (i) $G$ is a subgraph of $W_{n,k,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, \\ (ii) $k=2$, $c$ is odd, and $G$ is a subgraph of a member of $\mathcal{X}_{n,c}\cup \mathcal{Y}_{n,c}$, or \\ (iii) $k\geq 3$ and $G$ is a subgraph of the union of a clique $K_{c-k+1}$ and some cliques $K_{k+1}$'s, where any two cliques share the same two vertices. This provides a unified generalization of the above result of F\"{u}redi et al. as well as a recent result of Li et al. and independently, of F\"{u}redi et al. on non-Hamiltonian graphs. Moreover, we prove a stability result on a classical theorem of Bondy on the circumference. We use a novel approach, which combines several proof ideas including a closure operation and an edge-switching technique.