## Seminars and Colloquia by Series

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.
Thursday, September 14, 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 continue our discussion on the operations we use for characterizing feasible (G, a0, a1, a2, b1, b2). If time permits, we will also discuss useful structures for obtaining that characterization, such as frame, ideal frame, and framework. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.
Thursday, September 7, 2017 - 13:30 , Location: Skiles 005 , Shijie Xie , School of Mathematics, Georgia Tech , Organizer: Xingxing Yu
Let $G$ be a graph containing 5 different vertices $a_0, a_1, a_2, b_1$ and $b_2$. We say that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible if $G$ contains disjoint connected subgraphs $G_1, G_2$, such that  $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We give a characterization for $(G,a_0,a_1,a_2,b_1,b_2)$ to be feasible, answering a question of Robertson and Seymour. This is joint work with Changong Li, Robin Thomas, and Xingxing Yu.In this talk, we will discuss the operations we will use to reduce $(G,a_0,a_1,a_2,b_1,b_2)$ to $(G',a_0',a_1',a_2',b_1',b_2')$ with $|V(G)|+|E(G)|>|V(G')|+E(G')$, such that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible  iff $(G',a_0',a_1',a_2'b_1',b_2')$ is feasible.
Thursday, May 18, 2017 - 15:05 , Location: Skiles 005 , Daniel Kral , University of Warwick , Organizer: Robin Thomas
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.
Thursday, March 16, 2017 - 15:05 , Location: Skiles 005 , Rose McCarty , Math, GT , Organizer: Robin Thomas
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.
Thursday, February 9, 2017 - 15:05 , Location: Skiles 005 , Genghua Fan , Center for Discrete Mathematics, Fuzhou University , Organizer: Xingxing Yu
A spanning subgraph $F$ of a graph $G$ iscalled an even factor of $G$ if each vertex of $F$ has even degreeat least 2 in $F$. It was proved that if a graph $G$ has an evenfactor, then it has an even factor $F$ with $|E(F)|\geq {4\over7}(|E(G)| + 1)$, which is best possible. Recently, Cheng et al.extended the result by considering vertices of degree 2. They provedthat if a graph $G$ has an even factor, then it has an even factor$F$ with $|E(F)|\geq {4\over 7}(|E(G)| + 1)+{1\over 7}|V_2(G)|$,where $V_2(G)$ is the set of vertices of degree 2 in $G$. They alsogave examples showing that the second coefficient cannot be largerthan ${2\over 7}$ and conjectured that if a graph $G$ has an evenfactor, then it has an even factor $F$ with $|E(F)|\geq {4\over7}(|E(G)| + 1)+ {2\over 7}|V_2(G)|$. We note that the conjecture isfalse if $G$ is a triangle. We confirm the conjecture for all graphson at least 4 vertices. Moreover, if $|E(H)|\leq {4\over 7}(|E(G)| +1)+ {2\over 7}|V_2(G)|$ for every even factor $H$ of $G$, then everymaximum even factor of $G$ is a 2-factor in which each component isan even circuit.
Thursday, January 19, 2017 - 15:05 , Location: Skiles 005 , Dion Gijswijt , TU Delft , , Organizer: Esther Ezra
Consider the following solitaire game on a graph. Given a chip configuration on the node set V, a move consists of taking a subset U of nodes and sending one chip from U to V\U along each edge of the cut determined by U. A starting configuration is winning if for every node there exists a sequence of moves that allows us to place at least one chip on that node. The (divisorial) gonality of a graph is defined as the minimum number of chips in a winning configuration. This notion belongs to the Baker-Norine divisor theory on graphs and can be seen as a combinatorial analog of gonality for algebraic curves. In this talk we will show that the gonality is lower bounded by the tree-width and, if time permits, that the parameter is NP-hard to compute. We will conclude with some open problems.
Thursday, November 3, 2016 - 13:30 , Location: Skiles 005 , Chun-Hung Liu , Princeton University , Organizer: Robin Thomas
A tournament is a directed graph obtained by orienting each edge of a complete graph. A set of vertices D is a dominating set in a tournament if every vertex not in D is pointed by a vertex in D. A tournament H is a rebel if there exists k such that every H-free tournament has a dominating set of size at most k. Wu conjectured that every tournament is a rebel. This conjecture, if true, implies several other conjectures about tournaments. However, we will prove that Wu's conjecture is false in general and prove a necessary condition for being rebels. In addition, we will prove that every 2-colorable tournament and at least one non-2-colorable tournament are rebels. The later implies an open case of a conjecture of Berger, Choromanski, Chudnovsky, Fox, Loebl, Scott, Seymour and Thomasse about coloring tournaments. This work is joint with Maria Chudnovsky, Ringi Kim, Paul Seymour and Stephan Thomasse.
Thursday, October 13, 2016 - 13:30 , Location: Skiles 005 , Pavel Skums , Department of Computer Science, Georgia State University , Organizer: Robin Thomas
Lately there was a growing interest in studying self-similarity and fractal properties of graphs, which is largely inspired by applications in biology, sociology and chemistry. Such studies often employ statistical physics methods that borrow some ideas from graph theory and general topology, but are not intended to approach the problems under consideration in a rigorous mathematical way. To the best of our knowledge, a rigorous combinatorial theory that defines and studies graph-theoretical analogues of topological fractals still has not been developed. In this paper we introduce and study discrete analogues of Lebesgue and Hausdorff dimensions for graphs. It turned out that they are closely related to well-known graph characteristics such as rank dimension and Prague (or Nesetril-Rodl) dimension. It allowed us to formally define fractal graphs and establish fractality of some graph classes. We show, how Hausdorff dimension of graphs is related to their Kolmogorov complexity. We also demonstrate fruitfulness of this interdisciplinary approach by discover a novel property of general compact metric spaces using ideas from hypergraphs theory and by proving an estimation for Prague dimension of almost all graphs using methods from algorithmic information theory.
Thursday, September 29, 2016 - 13:30 , Location: Skiles 005 , Yan Wang , Math, GT , Organizer: Robin Thomas
Stiebitz showed that a graph with minimum degree s+t+1 can be decomposed into vertex disjoint subgraphs G_1 and G_2 such that G_1 has minimum degree at least s and G_2 has minimum degree at least t. Motivated by this result, Norin conjectured that a graph with average degree s+t+2 can be decomposed into vertex disjoint subgraphs G_1 and G_2 such that G_1 has average degree at least s and G_2 has average degree at least t. Recently, we prove that a graph with average degree s+t+2 contains vertex disjoint subgraphs G_1 and G_2 such that G_1 has average degree at least s and G_2 has average degree at least t. In this talk, I will discuss the proof technique. This is joint work with Hehui Wu.