Crystal constructions in Number Theory

Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics

Perfect matchings: Modified Aztec diamonds, covering graphs andn-matchings

In the Introduction, we present the problems we are going to study and we establish the basic definitions, concepts and results that are used throughout. We begin the first chapter with a presentation of the Aztec diamond and the behaviour of its random domino tilings. We introduce the dual-matching-problem and we explore the structure of the perfect matchings of modified Aztec diamonds. We show that some of these matchings can be extended to matchings of the dual Aztec diamond, pointing out a bijection between these types of matchings. We determine the number of perfect matchings for each of the modified graphs and the placement probabilities of the edges belonging to such a matching at a given location. We conclude with a theorem presenting the common asymptotic behaviour of the dual and the modified Aztec diamonds and we deduce a version of the Arctic Circle Theorem for these graphs. The second part is dedicated to the study of non-ramified perfect n-matchings, their decomposition into perfect matchings and 2-matchings as well as their relations to the perfect matchings of covering graphs. For the n-covering graphs we use the permutation derived graph construction. We determine the number of liftings of a given n-matching to a matching of a branched covering graph and then of a n-covering graph, together with necessary and sufficient conditions for the existence of the lifting. In particular, for the case of 2-matchings, we obtain a uniform behaviour of liftings of cycles. First, we deduce a theorem that relates the number of perfect matchings of the branched covering graph we have introduced to the number of perfect 2-matchings of the initial graph. Then we study the 2-covering graphs, their number, we determine the number of liftings of a 2-matchings (as a power of 2) and we obtain a theorem that characterizes the 2-matchings as the average of perfect matchings of 2-covering graphs. We conclude with some considerations about the maximum, minimum and the realization of this average and methods of computing it

Maximum Wiener Indices of Unicyclic Graphs of Given Matching Number

In this article, we determine the maximum Wiener indices of unicyclic graphs with given number of vertices and matching number. We also characterize the extremal graphs. This solves an open problem of Du and Zhou.Comment: 14 pages, 9 figure

The number of maximum matchings in a tree

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order $n$ is at most $O(1.391664^n)$ (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by G\'orska and Skupie\'n on the number of maximal matchings (maximal with respect to set inclusion).Comment: 38 page

Common graphs with arbitrary chromatic number

Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of K_k, and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko's Conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common, and in fact, no 2-chromatic common graph unsettled for Sidorenko's Conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known. We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami, Hladky, Kral, Norine and Razborov [Combin. Probab. Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov [London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger, Stovicek and Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common graph with chromatic number at least four.Comment: Updated to include reference to arXiv:2207.0942

On Structures of Large Rooted Graphs

A rooted graph is a pair (G,R), where G is a graph and R⊆V(G). There are two research topics in this thesis. One is about unavoidable substructures in sufficiently large rooted graphs. The other is about characterizations of rooted graphs excluding specific large graphs. The first topic of this thesis is motivated by Ramsey Theorem, which states that K_n and ¯(K_n ) are unavoidable induced subgraphs in every sufficiently large graph. It is also motivated by a classical result of Oporowski, Oxley, and Thomas, which determines unavoidable large 3-connected minors. We first determine unavoidable induced subgraphs, and unavoidable subgraphs in connected graphs with sufficiently many roots. We also extend this result to generalized rooted connected graphs. Secondly, we extend these results to rooted graphs of higher connectivity. In particular, we determine unavoidable subgraphs of sufficiently large rooted 2- connected graphs. Again, this result is extended to generalized rooted 2-connected graphs. The second topic of this dissertation is motivated by two results of Robertson and Seymour, let’s only talk about path and star. In the first result they established that graphs without a long path subgraph are precisely those that can be constructed using a specific operation within a bounded number of iterations, starting from the trivial graph. In the second result they showed that graphs without a large star minor are those that are subdivisions of graphs with bounded number vertices. We consider similar problems for path, star and comb. We have some theorems on characterizations of rooted connected graphs excluding a heavy path, a large (nicely) confined comb, a large (nicely) confined star, which are similar to those of Robertson and Seymour. Moreover, our results strengthen their related results