Cycle Double Covers and Integer Flows

My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems

Reconfiguration of basis pairs in regular matroids

In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns the exchange distance of two matroid basis sequences, a problem that appears in several areas of computer science and mathematics. In 1980, White proposed a conjecture for the characterization of two basis sequences being reachable from each other by symmetric exchanges, which received a significant interest also in algebra due to its connection to toric ideals and Gr\"obner bases. In this work, we verify White's conjecture for basis sequences of length two in regular matroids, a problem that was formulated as a separate question by Farber, Richter, and Shan and Andres, Hochst\"attler, and Merkel. Most of previous work on White's conjecture has not considered the question from an algorithmic perspective. We study the problem from an optimization point of view: our proof implies a polynomial algorithm for determining a sequence of symmetric exchanges that transforms a basis pair into another, thus providing the first polynomial upper bound on the exchange distance of basis pairs in regular matroids. As a byproduct, we verify a conjecture of Gabow from 1976 on the serial symmetric exchange property of matroids for the regular case.Comment: 28 pages, 6 figure

Shortest k-Disjoint Paths via Determinants

The well-known $k$-disjoint path problem ($k$-DPP) asks for pairwise vertex-disjoint paths between $k$ specified pairs of vertices $(s_i, t_i)$ in a given graph, if they exist. The decision version of the shortest $k$-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest $k$-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdiere and Schrijver for the general one-face problem. We do so by providing a randomised $NC^2$ algorithm along with an $O(n^{\omega})$ time randomised sequential algorithm. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals. In addition, the earlier best known sequential algorithms (e.g. Borradaile et al.) were randomised while ours are also deterministic. The algorithms are based on a bijection between a shortest $k$-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "good" cycle covers.Comment: 17 pages, 6 figure

Establishing a Connection Between Graph Structure, Logic, and Language Theory

The field of graph structure theory was given life by the Graph Minors Project of Robertson and Seymour, which developed many tools for understanding the way graphs relate to each other and culminated in the proof of the Graph Minors Theorem. One area of ongoing research in the field is attempting to strengthen the Graph Minors Theorem to sets of graphs, and sets of sets of graphs, and so on. At the same time, there is growing interest in the applications of logic and formal languages to graph theory, and a significant amount of work in this field has recently been consolidated in the publication of a book by Courcelle and Engelfriet. We investigate the potential applications of logic and formal languages to the field of graph structure theory, suggesting a new area of research which may provide fruitful

Results in Extremal Graph Theory, Ramsey Theory and Additive Combinatorics

This dissertation contains results from various areas of Combinatorics. In Chapter 2, we consider a central problem in Extremal Graph Theory. The extremal number (or Turán number) ex(n,H) of a graph H is the maximum number of edges in an H-free graph on n vertices. It is a major area of research to better understand the extremal number of bipartite graphs. In this chapter we develop a new method which allows us to obtain strong (and often best possible) upper bounds in a wide range of cases. Our results answer several conjectures of Conlon and Lee, and Kang, Kim and Liu. Furthermore, they relate to and improve work of (among others) Füredi, Alon, Krivelevich and Sudakov, Kostochka and Pyber, and Jiang and Seiver. While in Chapter 2 the focus is on subdivided graphs, in Chapter 3 we study the extremal number of blow-ups. In particular, we obtain tight upper bounds for the extremal number of blow-ups of trees. As an extension of this, we pose a general conjecture relating the extremal number of F and that of its blow-up. We prove the conjecture for the 2-blowup of C_6. In Chapter 4 we study a coloured variant of the Turán problem. The rainbow Turán number of H, denoted by ex*(n,H), is the maximum possible number of edges in an n-vertex properly edge-coloured graph without a rainbow subgraph isomorphic to H. We prove that ex*(n,C_{2k})=O(n^{1+1/k}), which is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraete. We use the same method to answer several further questions in various topics: among others, a question of Conlon and Tyomkyn on colour-isomorphic cycles and a conjecture of Jiang and Newman of blow-ups of cycles. We also disprove an old conjecture of Erdős and Simonovits on (ordinary) extremal numbers. In Chapter 5, we consider the following problem. Let 2 ≤ s < t be fixed integers. If G is an arbitrary K_t-free graph on n vertices, how large a K_s-free induced subgraph must there exist in G? This number, which is a generalisation of the usual off-diagonal Ramsey numbers, is viewed as a function in n, and is called the Erdős-Rogers function. We obtain new upper bounds in the case s+2 ≤ t ≤ 2s-1, improving results of (among others) Bollobás, Erdős and Krivelevich, and answering a question of Dudek, Retter and Rödl. In Chapter 6, we investigate the relationship between two well-studied notions of tensor rank. We show that the partition rank of a tensor is bounded above by a polynomial in the analytic rank of the same tensor. This improves Ackermann-type bounds obtained by various authors including Green and Tao, and Bhowmick and Lovett. In Chapter 7, we use the main technical lemma of Chapter 6 to prove a result about the expansion of subsets of the Cayley graph on the tensor product F_2^{n_1} ⊗ … ⊗ F_2^{n_d} where the generators are the rank 1 tensors. This is motivated by the famous Unique Games Conjecture from Theoretical Computer Science, and is a partial generalisation of a recent breakthrough result of Khot, Minzer and Safra. In Chapter 8, we ask the following question. Given constants α,β,γ, what is the minimal possible edge density of a graph G on n vertices with the property that every subset A⊆V(G) with |A| ≥ αn contains a subset B⊆A with |B| ≥ βn such that G[B] has edge density at least γ? We also study a bipartite version of this question, obtaining sharp results in both cases. In Chapter 9, we determine asymptotically the maximum possible number of induced C_5's in a planar graph on n vertices

Asymptotics of lowest unitary SL(2,C) invariants on graphs

We describe a technique to study the asymptotics of SL(2,C) invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski, Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known asymptotics of the 4-simplex graph with a different perspective on the geometric variables and introduce an algorithm valid for any graph. On general grounds, we find that critical configurations are not just Regge geometries, but a larger set corresponding to conformal twisted geometries. These can be either Euclidean or Lorentzian, and include curved and flat 4d polytopes as subsets. For modular graphs, we show that multiple pairs of critical points exist, and there exist critical configurations of mixed signature, Euclidean and Lorentzian in different subgraphs, with no 4d embedding possible.Comment: 40 Pages + 5 Appendices. 11 Figures. v2: Refined presentation of the general algorithm, additional minor amendments. v3: paragraph added in section 5 about curved embedding