A 1-2-3-4 result for the 1-2-3 Conjecture in 5-regular graphs

International audienceThe 1-2-3 Conjecture, posed by Karoński, Łuczak and Thomason, asks whether every connected graph G different from K2 can be 3-edge-weighted so that every two adjacent vertices of G get distinct sums of incident weights. Towards that conjecture, the best-known result to date is due to Kalkowski, Karoński and Pfender, who proved that it holds when relaxed to 5-edge-weightings. Their proof builds upon a weighting algorithm designed by Kalkowski for a total version of the problem. In this work, we present new mechanisms for using Kalkowski's algorithm in the context of the 1-2-3 Conjecture. As a main result we prove that every 5-regular graph admits a 4-edge-weighting that permits to distinguish its adjacent vertices via their incident sums

On matchings and factors of graphs /

In Section 1, we recall the historical sketch of matching and factor theory of graphs, and also introduce some necessary definitions and notation. In Section 2, we present a sufficient condition for the existence of a (g, f)-factor in graphs with the odd-cycle property, which is simpler than that of Lovasz\u27s (g, f)-Factor Theorem. From this, we derive some further results, and we show that (a) every r-regular graph G with the odd-cycle property has a k-factor, where 0 ≤ k ≤ r and k|V(G)| ≡ 0 (mod 2), (b) every graph G with the strong odd-cycle property with k|V(G)|≡ 0 (mod 2) is k-factorable if and only if G is a km-regular graph for some m ≥ 1, and (c) every regular graph of even order with the strong odd-cycle property is of the second class (i.e. the edge chromatic number is Δ). Chvátal [26] presented the following two conjectures that (1) a graph G has a 2-factor if tough(G) ≥ 3/2, and (2) a graph G has a k-factor if k|V(G)| ≡ 0 (mod 2) and tough(G) ≥ k. Enomoto et.al. [32] proved the second conjecture. They also proved the sharpness of the bound on tough(G) that guarantees the existence of a k-factor. This implies that the first conjecture is false. In Section 3, we show that the result of the second conjecture can be improved in some sense, and the first conjecture is also true if the graph considered has the odd-cycle property. Anderson [3] stated that a graph G of even order has a 1-factor if bind(G) ≥ 4/3, and Katerinis and Woodall [48] proved that a graph G of order n has a k-factor if bind(G) ˃ (2k -I)(n - 1)/(k(n - 2) + 3), where k ≥ 2, n ≥ 4k - 6 and kn ≡ 0 (mod 2). In Section 4, we shall present some similar conditions for the existence of [a, b]-factors. In Section 5, we study the existence of [a, b]-parity-factors in a graph, among which we extend some known theorems from 1-factors to {1, 3, ... , 2n - 1}-factors, or from k-factors to [a, b]-parity-factors. Also, extending Petersen\u27s 2-Factorization Theorem, we proved that a graph is [2a, 2b]-even-factorable if and only if it is a [2na, 2nb]-even-graph for some n ≥ 1. Plummer showed that (a) (in [58]) every graph G of even order is k-extendable if tough(G) ˃ k, and (b) (in [59]) every (2k+1)-connected graph G is k-extendable if G is K1,3-free, respectively. In Section 6, we give a counterpart of the former in terms of binding number, and extend the latter from K1,3-free graphs to K1,n-free graphs. Furthermore, we present a result toward the problem, posed by Saito [61] and Plummer [60], of characterizing the graphs that are maximal k-extendable

Connectivity and spanning trees of graphs

This dissertation focuses on connectivity, edge connectivity and edge-disjoint spanning trees in graphs and hypergraphs from the following aspects.;1. Eigenvalue aspect. Let lambda2(G) and tau( G) denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of tau(G), Cioaba and Wong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if lambda 2(G)) \u3c d -- 2k-1d+1 , then tau(G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. We propose a more general conjecture that for a graph G with minimum degree delta ≥ 2 k ≥ 4, if lambda2(G) \u3c delta -- 2k-1d+1 then tau(G) ≥ k. We prove the conjecture for k = 2, 3 and provide partial results for k ≥ 4. We also prove that for a graph G with minimum degree delta ≥ k ≥ 2, if lambda2( G) \u3c delta -- 2k-1d +1 , then the edge connectivity is at least k. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on tau(G) and edge connectivity.;2. Network reliability aspect. With graphs considered as natural models for many network design problems, edge connectivity kappa\u27(G) and maximum number of edge-disjoint spanning trees tau(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G. Let kappa\u27(G) = max{lcub}kappa\u27(H) : H is a subgraph of G{rcub}. We present: (i) For each integer k \u3e 0, a characterization for graphs G with the property that kappa\u27(G) ≤ k but for any additional edge e not in G, kappa\u27(G + e) ≥ k + 1. (ii) For any integer n \u3e 0, a characterization for graphs G with |V(G)| = n such that kappa\u27(G) = tau( G) with |E(G)| minimized.;3. Generalized connectivity. For an integer l ≥ 2, the l-connectivity kappal( G) of a graph G is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Let k ≥ 1, a graph G is called (k, l)-connected if kappa l(G) ≥ k. A graph G is called minimally (k, l)-connected if kappal(G) ≥ k but ∀e ∈ E( G), kappal(G -- e) ≤ k -- 1. A structural characterization for minimally (2, l)-connected graphs and some extremal results are obtained. These extend former results by Dirac and Plummer on minimally 2-connected graphs.;4. Degree sequence aspect. An integral sequence d = (d1, d2, ···, dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r- uniform hypergraph with degree sequence d. It is proved that an r-uniform hypergraphic sequence d = (d1, d2, ···, dn) has a k-edge-connected realization if and only if both di ≥ k for i = 1, 2, ···, n and i=1ndi≥ rn-1r-1 , which generalizes the formal result of Edmonds for graphs and that of Boonyasombat for hypergraphs.;5. Partition connectivity augmentation and preservation. Let k be a positive integer. A hypergraph H is k-partition-connected if for every partition P of V(H), there are at least k(| P| -- 1) hyperedges intersecting at least two classes of P. We determine the minimum number of hyperedges in a hypergraph whose addition makes the resulting hypergraph k-partition-connected. We also characterize the hyperedges of a k-partition-connected hypergraph whose removal will preserve k-partition-connectedness

Extremal problems on special graph colorings

In this thesis, we study several extremal problems on graph colorings. In particular, we study monochromatic connected matchings, paths, and cycles in 2-edge colored graphs, packing colorings of subcubic graphs, and directed intersection number of digraphs. In Chapter 2, we consider monochromatic structures in 2-edge colored graphs. A matching M in a graph G is connected if all the edges of M are in the same component of G. Following Łuczak, there are a number of results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 2-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings, which is used to find necessary and sufficient conditions on the existence of monochromatic paths and cycles: for every fixed s and large n, we describe all values of n_1, ...,n_s such that for every 2-edge-coloring of the complete s-partite graph K_{n_1, ...,n_s} there exists a monochromatic (i) cycle C_{2n} with 2n vertices, (ii) cycle C_{at least 2n} with at least 2n vertices, (iii) path P_{2n} with 2n vertices, and (iv) path P_{2n+1} with 2n+1 vertices. Our results also imply for large n of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every 2-edge-coloring of the complete 3-partite graph K_{n,n,n} there is a monochromatic path P_{2n+1}. Moreover, we prove that for every sufficiently large n, if n = 3t+r where r in {0,1,2} and G is an n-vertex graph with minimum degree at least (3n-1)/4, then for every 2-edge-coloring of G, either there are cycles of every length {3, 4, 5, ..., 2t+r} of the same color, or there are cycles of every even length {4, 6, 8, ..., 2t+2} of the same color. This result is tight and implies the conjecture of Schelp that for every sufficiently large n, every (3n-1)-vertex graph G with minimum degree larger than 3|V(G)|/4, in each 2-edge-coloring of G there exists a monochromatic path P_{2n} with 2n vertices. It also implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for every positive integer n of the form n=3t+r where r in {0,1,2} and every n-vertex graph G with minimum degree at least 3n/4, in each 2-edge-coloring of G there exists a monochromatic cycle of length at least 2t+r. In Chapter 3, we consider a collection of special vertex colorings called packing colorings. For a sequence of non-decreasing positive integers S = (s_1, ..., s_k), a packing S-coloring is a partition of V(G) into sets V_1, ..., V_k such that for each integer i in {1, ..., k} the distance between any two distinct x,y in V_i is at least s_i+1. The smallest k such that G has a packing (1,2, ..., k)-coloring is called the packing chromatic number of G and is denoted by \chi_p(G). The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We show that for every fixed k and g at least 2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k, which answers the previous question in the negative. Moreover, we work towards the conjecture of Brešar, Klavžar, Rall and Wash that the packing chromatic number of 1-subdivision of subcubic graphs are bounded above by 5. In particular, we show that every subcubic graph is (1,1,2,2,3,3,k)-colorable for every integer k at least 4 via a coloring in which color k is used at most once, every 2-degenerate subcubic graph is (1,1,2,2,3,3)-colorable, and every subcubic graph with maximum average degree less than 30/11 is packing (1,1,2,2)-colorable. Furthermore, while proving the packing chromatic number of subcubic graphs is unbounded, we also consider improving upper bound on the independence ratio, alpha(G)/n, of cubic n-vertex graphs of large girth. We show that ``almost all" cubic labeled graphs of girth at least 16 have independence ratio at most 0.454. In Chapter 4, we introduce and study the directed intersection representation of digraphs. A directed intersection representation is an assignment of a color set to each vertex in a digraph such that two vertices form an edge if and only if their color sets share at least one color and the tail vertex has a strictly smaller color set than the head. The smallest possible size of the union of the color sets is defined to be the directed intersection number (DIN). We show that the directed intersection representation is well-defined for all directed acyclic graphs and the maximum DIN among all n vertex acyclic digraphs is at most 5n^2/8 + O(n) and at least 9n^2/16 + O(n)

Well-Distributed Sequences: Number Theory, Optimal Transport, and Potential Theory

The purpose of this dissertation will be to examine various ways of measuring how uniformly distributed a sequence of points on compact manifolds and finite combinatorial graphs can be, providing bounds and novel explicit algorithms to pick extremely uniform points, as well as connecting disparate branches of mathematics such as Number Theory and Optimal Transport. Chapter 1 sets the stage by introducing some of the fundamental ideas and results that will be used consistently throughout the thesis: we develop and establish Weyl\u27s Theorem, the definition of discrepancy, LeVeque\u27s Inequality, the Erdős-Turán Inequality, Koksma-Hlawka Inequality, and Schmidt\u27s Theorem about Irregularities of Distribution. Chapter 2 introduces the Monge-Kantorovich transport problem with special emphasis on the Benamou-Brenier Formula (from 2000) and Peyre\u27s inequality (from 2018). Chapter 3 explores Peyre\u27s Inequality in further depth, considering how specific bounds on the Wasserstein distance between a point measure and the uniform measure may be obtained using it, in particular in terms of the Green\u27s function of the Laplacian on a manifold. We also show how a smoothing procedure can be applied by propagating the heat equation on probability mass in order to get stronger bounds on transport distance using well-known properties of the heat equation. In Chapter 4, we turn to the primary question of the thesis: how to select points on a space which are as uniformly distributed as possible. We consider various diverse approaches one might attempt: an ergodic approach iterating functions with good mixing properties; a dyadic approach introduced in a 1975 theorem of Kakutani on proportional splittings on intervals; and a completely novel potential theoretic approach, assigning energy to point configurations and greedily minimizing the total potential arising from pair-wise point interactions. Such energy minimization questions are certainly not new, in the static setting--physicist Thomson posed the question of how to minimize the potential of electrons on a sphere as far back as 1904. However, a greedy approach to uniform distribution via energy minimization is novel, particularly through the lens of Wasserstein, and yields provably Wasserstein-optimal point sequences using the Green\u27s function of the Laplacian as our energy function on manifolds of dimension at least 3 (with dimension 2 losing at most a square root log factor from the optimal bound). We connect this to known results from Graham, Pausinger, and Proinov regarding best possible uniform bounds on the Wasserstein 2-distance of point sequences in the unit interval. We also present many open questions and conjectures on the optimal asymptotic bounds for total energy of point configurations and the growth of the total energy function as points are added, motivated by numerical investigations that display remarkably well-behaved qualities in the dynamical system induced by greedy minimization. In Chapter 5, we consider specific point sequences and bounds on the transport distance from the point measure they generate to the uniform measure. We provide provably optimal rates for the van der Corput sequence, the Kronecker sequence, regular grids and the measures induced by quadratic residues in a field of prime order. We also prove an upper bound for higher degree monomial residues in fields of prime order, and conjecture this to be optimal. In Chapter 6, we consider numerical integration error bounds over Lipschitz functions, asking how closely we can estimate the integral of a function by averaging its values at finitely many points. This is a rather classical question that was answered completely by Bakhalov in 1959 and has since become a standard example (`the easiest case which is perfectly understood\u27). Somewhat surprisingly perhaps, we show that the result is not sharp and improve it in two ways: by refining the function space and by proving that these results can be true uniformly along a subsequence. These bounds refine existing results that were widely considered to be optimal, and we show the intimate connection between transport distance and integration error. Our results are new even for the classical discrete grid. In Chapter 7, we study the case of finite graphs--we show that the fundamental question underlying this thesis can also be meaningfully posed on finite graphs where it leads to a fascinating combinatorial problem. We show that the philosophy introduced in Chapter 4 can be meaningfully adapted and obtain a potential-theoretic algorithm that produces such a sequence on graphs. We show that, using spectral techniques, we are able to obtain empirically strong bounds on the 1-Wasserstein distance between measures on subsets of vertices and the uniform measure, which for graphs of large diameter are much stronger than the trivial diameter bound

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio