We discuss flows (and group-connectivity) in signed graphs, and prove a new result about group-connectivity in 3-edge-connected signed graphs. This is joint work with Alejandra Brewer Castano and Kathryn Nurse.
Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either a complete graph on r vertices or an independent set with r vertices. It is well known that every sufficiently large, connected graph contains an induced subgraph isomorphic to one of a large complete graph, a large star, and a long path. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray. We also prove an analogous result for infinite 2-connected graphs.
We show that if each edge of the complete bipartite graph $K_{n,n}$ is given a random list of $C(\log n)$ colors from $[n]$, then with high probability, there is a proper edge coloring where the color of each edge comes from the corresponding list. We also prove analogous results for Latin squares and Steiner triple systems. This resolves several related conjectures of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus. I will discuss some of the main ingredients which go into the proof: the Kahn-Kalai conjecture, absorption, and the Lovasz Local Lemma distribution. Based on joint work with Huy Tuan Pham.
Many results in extremal graph theory build on supersaturation of subgraphs. In other words, when a graph is dense enough, it contains many copies of a certain subgraph and these copies are then used as building blocks to force another subgraph of interest. Recently more success is found within this approach where one utilizes not only the large number of copies of a certain subgraph but a well-distributed collection of them to force the desired subgraph. We discuss some recent progress of this nature. The talk is built on joint work with Sean Longbrake, and with Sean Longbrake and Jie Ma.
In this talk we discuss our proof of a recent conjecture of Guo and Poznanovi\'{c} concerning chains in certain 01-fillings of moon polyominoes. A key ingredient of our proof is a correspondence between words $w$ and pairs $(\mathcal{W}(w), \mathcal{M}(w))$ of increasing tableaux such that $\mathcal{M}(w)$ determines the lengths of the longest strictly increasing and strictly decreasing sequences in every subinterval of $w$. (It will be noted that similar and well-studied correspondences like RSK insertion and Hecke insertion fail in this regard.) To define our correspondence we make use of Thomas and Yong's K-infusion operator and then use it to obtain the bijections that prove the conjecture of Guo and Poznanovi\'{c}. (Joint work with D. Saracino.)
The edge-coloring problem (ECP) for a multigraph $G=(V, E)$ is to color its edges with minimum number of colors such that no two adjacent vertices receive the same color. ECP can be naturally formulated as an integer program, and its linear programming relaxation is referred to as the fractional edge-coloring problem (FECP). The optimal value of ECP (resp. FECP) is called the chromatic index (resp. fractional chromatic index) of $G$, denoted by $\chi^{\prime}(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let $ \mathcal{W}^*(G) $ be the fractional density of $G$, defined by $$ \mathcal{W}^*(G) = \max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor|U|/2\rfloor}. $$ Seymour showed that $\chi^*(G)=\max \{\Delta(G), \mathcal{W}^*(G)\}$. Moreover, the Goldberg-Seymour Conjecture is confirmed Chen, Jing, and Zang states that $\chi^{\prime}(G) \leq \max \{\Delta(G)+1,\lceil\mathcal{W}^*(G)\rceil\}$. Chen, Zang and Zhao developed an algorithm that calculates $ \mathcal{W}^*(G) $ in strongly polynomial time. Inspired by their results, we consider the fractional $ f $-edge-coloring problem ($ f $-FECP) for a given function $ f:V\to \mathbb N_+ $, which is a generalization of FECP: each spanning subgraph induced by a color class has degree at most $ f(v) $ at each vertex $ v\in V $. We give a strongly polynomial-time algorithm for calculating the corresponding fractional $ f $-density $$ \mathcal{W}^*_{f}(G)=\max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor f(U) / 2\rfloor}. $$
When can we find perfect matchings in hypergraphs whose vertices represent group elements and whose edges represent solutions to systems of linear equations? A prototypical problem of this type is the Hall-Paige conjecture, which asks for a characterisation of the groups whose multiplication table (viewed as a Latin square) contains a transversal. Other problems expressible in this language include the toroidal n-queens problem, Graham-Sloane harmonious tree-labelling conjecture, Ringel's sequenceability conjecture, Snevily's subsquare conjecture, Tannenbaum's zero-sum conjecture, and many others. All of these problems have a similar flavour, yet until recently they have been approached in completely different ways, using algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. In this talk we discuss a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we will see that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups. Joint work with Alexey Pokrosvkiy
An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog n (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of "random-like’’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables.
The proof proceeds via an "additive structure’’ dichotomy on the degree sequence, and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics, and low-rank approximation. One of the consequences of our result is the resolution of an old conjecture of Erdos and McKay, for which he offered one of his notorious monetary prizes.
(Joint work with Matthew Kwan, Ashwin Sah and Lisa Sauermann)
The chromatic index $\chi'(G)$ of a graph $G$ is the least number of colors assigned to the edges of $G$ such that no two adjacent edges receive the same color. The total chromatic number $\chi''(G)$ of a graph $G$ is the least number of colors assigned to the edges and vertices of $G$ such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color as its two endpoints. The chromatic index and the total chromatic number are two of few fundamental graph parameters, and their correlation has always been a subject of intensive study in graph theory.
By definition, $\chi'(G) \le \chi''(G)$ for every graph $G$. In 1984, Goldberg conjectured that for any multigraph $G$, if $\chi'(G) \ge \Delta(G) +3$ then $\chi''(G) = \chi'(G)$. We show that Goldberg's conjecture is asymptotically true. More specifically, we prove that for a multigraph $G$ with maximum degree $\Delta$ sufficiently large, $\chi''(G) = \chi'(G)$ provided $\chi'(G) \ge \Delta + 10\Delta^{35/36}$. When $\chi'(G) \ge \Delta(G) +2$, the chromatic index $\chi'(G)$ is completely determined by the fractional chromatic index. Consequently, the total chromatic number $\chi''(G)$ can be computed in polynomial-time in this case.
The Matroid Minors Project of Geelen, Gerards, and Whittle describes the structure of minor-closed classes of matroids representable by a matrix over a fixed finite field. To use these results to study specific classes, it is important to study the matroids in the class containing spanning cliques. A spanning clique of a matroid M is a complete-graphic restriction of M with the same rank as M.
In this talk, we will describe the structure of dyadic matroids with spanning cliques. The dyadic matroids are those matroids that can be represented by a real matrix each of whose nonzero subdeterminants is a power of 2, up to a sign. A subclass of the dyadic matroids is the signed-graphic matroids. In the class of signed-graphic matroids, the entries of the matrix are determined by a signed graph. Our result is that dyadic matroids with spanning cliques are signed-graphic matroids and a few exceptional cases.
The main results in this talk will come from joint work with Ben Clark, James Oxley, and Stefan van Zwam. This talk will include a brief introduction to matroids.
