44,628 research outputs found

    A Strong Edge-Coloring of Graphs with Maximum Degree 4 Using 22 Colors

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    In 1985, Erd\H{o}s and Ne\'{s}etril conjectured that the strong edge-coloring number of a graph is bounded above by 5/4Δ2{5/4}\Delta^2 when Δ\Delta is even and 1/4(5Δ2−2Δ+1){1/4}(5\Delta^2-2\Delta+1) when Δ\Delta is odd. They gave a simple construction which requires this many colors. The conjecture has been verified for Δ≀3\Delta\leq 3. For Δ=4\Delta=4, the conjectured bound is 20. Previously, the best known upper bound was 23 due to Horak. In this paper we give an algorithm that uses at most 22 colors.Comment: 9 pages, 4 figure

    Extremal results in sparse pseudorandom graphs

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    Szemer\'edi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemer\'edi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.Comment: 70 pages, accepted for publication in Adv. Mat

    Toric algebra of hypergraphs

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    The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in algebraic statistics and to combinatorial discrepancy. Section 6 (open problems) has been moderately revise

    Improved Analysis of Deterministic Load-Balancing Schemes

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    We consider the problem of deterministic load balancing of tokens in the discrete model. A set of nn processors is connected into a dd-regular undirected network. In every time step, each processor exchanges some of its tokens with each of its neighbors in the network. The goal is to minimize the discrepancy between the number of tokens on the most-loaded and the least-loaded processor as quickly as possible. Rabani et al. (1998) present a general technique for the analysis of a wide class of discrete load balancing algorithms. Their approach is to characterize the deviation between the actual loads of a discrete balancing algorithm with the distribution generated by a related Markov chain. The Markov chain can also be regarded as the underlying model of a continuous diffusion algorithm. Rabani et al. showed that after time T=O(log⁥(Kn)/ÎŒ)T = O(\log (Kn)/\mu), any algorithm of their class achieves a discrepancy of O(dlog⁥n/ÎŒ)O(d\log n/\mu), where ÎŒ\mu is the spectral gap of the transition matrix of the graph, and KK is the initial load discrepancy in the system. In this work we identify some natural additional conditions on deterministic balancing algorithms, resulting in a class of algorithms reaching a smaller discrepancy. This class contains well-known algorithms, eg., the Rotor-Router. Specifically, we introduce the notion of cumulatively fair load-balancing algorithms where in any interval of consecutive time steps, the total number of tokens sent out over an edge by a node is the same (up to constants) for all adjacent edges. We prove that algorithms which are cumulatively fair and where every node retains a sufficient part of its load in each step, achieve a discrepancy of O(min⁥{dlog⁥n/ÎŒ,dn})O(\min\{d\sqrt{\log n/\mu},d\sqrt{n}\}) in time O(T)O(T). We also show that in general neither of these assumptions may be omitted without increasing discrepancy. We then show by a combinatorial potential reduction argument that any cumulatively fair scheme satisfying some additional assumptions achieves a discrepancy of O(d)O(d) almost as quickly as the continuous diffusion process. This positive result applies to some of the simplest and most natural discrete load balancing schemes.Comment: minor corrections; updated literature overvie

    On largest volume simplices and sub-determinants

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    We show that the problem of finding the simplex of largest volume in the convex hull of nn points in Qd\mathbb{Q}^d can be approximated with a factor of O(log⁡d)d/2O(\log d)^{d/2} in polynomial time. This improves upon the previously best known approximation guarantee of d(d−1)/2d^{(d-1)/2} by Khachiyan. On the other hand, we show that there exists a constant c>1c>1 such that this problem cannot be approximated with a factor of cdc^d, unless P=NPP=NP. % This improves over the 1.091.09 inapproximability that was previously known. Our hardness result holds even if n=O(d)n = O(d), in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where cˉ\bar c is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d×nd\times n matrix

    Functional limit theorems for random regular graphs

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    Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemer\'edi argument for estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and Related Field
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