1,123 research outputs found

    On the spectrum of hypergraphs

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    Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different structural properties of a hypergrpah, can be well studied using spectral properties of these matrices. Connectivity of a hypergraph is also investigated by the eigenvalues of these operators. Spectral radii of the same are bounded by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by the eigenvalues of its connectivity matrices. We characterize different properties of a regular hypergraph characterized by the spectrum. Strong (vertex) chromatic number of a hypergraph is bounded by the eigenvalues. Cheeger constant on a hypergraph is defined and we show that it can be bounded by the smallest nontrivial eigenvalues of Laplacian matrix and normalized Laplacian matrix, respectively, of a connected hypergraph. We also show an approach to study random walk on a (non-uniform) hypergraph that can be performed by analyzing the spectrum of transition probability operator which is defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two different ways. We show that if the Laplace operator, Δ\Delta, on a hypergraph satisfies a curvature-dimension type inequality CD(m,K)CD (\mathbf{m}, \mathbf{K}) with m>1\mathbf{m}>1 and K>0\mathbf{K}>0 then any non-zero eigenvalue of Δ- \Delta can be bounded below by mKm1 \frac{ \mathbf{m} \mathbf{K}}{ \mathbf{m} -1 } . Eigenvalues of a normalized Laplacian operator defined on a connected hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph

    Embedding large subgraphs into dense graphs

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    What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

    Decompositions into subgraphs of small diameter

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    We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E) on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that |E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma, Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a constant depending only \epsilon. This shows that every dense graph can be partitioned into a small number of ``small worlds'' provided that few edges can be ignored. Improving on their result, we determine P(n,\epsilon,d) within an absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded for \epsilon n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, R\"odl, Ruci\'nski, and Szemer\'edi.Comment: 18 page

    Distributed Symmetry Breaking in Hypergraphs

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    Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and Δ+1\Delta +1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in O(log2n)O(\log^2 n) rounds (nn is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an O(Δϵpolylog(n))O(\Delta^{\epsilon}\text{polylog}(n))-round hypergraph MIS algorithm in the CONGEST model where Δ\Delta is the maximum node degree of the hypergraph and ϵ>0\epsilon > 0 is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde
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