115,757 research outputs found

    Decomposing Complete Graphs into a Graph Pair of Order 6

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    Firstly, a graph G consists of a vertex set V (G), and an edge set E (G) of endpoints which relate two vertices with each edge. Also, a decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list. In the field of graph theory, graph decomposition is an active field of research. A graph pair is a pair of graphs on the same vertex set whose union is the complete graph. Abueida and Daven studied decompositions of complete graphs into graph-pairs of order four and five. We are extending their results by investigating which complete graphs decompose into a specific graph pair of order 6

    A Look at Multi-Decompositions of Complete Graphs into Graph Pairs of Order 4

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    Firstly, a graph G consists of a vertex set V (G), and an edge set E (G) of endpoints which relate two vertices with each edge. Also, a decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list. In the field ofgraph theory, graph decomposition is an active field of research. One type of decomposition is graph pairs. A graph pair is a pair of graphs on the same vertex set whose union is the complete graph. Abueida and Daven studieddecompositions of complete graphs into graph-pairs of order four. In their proof, they left a small part to the readers. We will complete this proof

    Towards obtaining a 3-Decomposition from a perfect Matching

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    A decomposition of a graph is a set of subgraphs whose edges partition those of GG. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list. In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching MM from a cubic graph GG leaves a disjoint union of cycles. Contracting these cycles yields a new graph GMG_M. The graph GG is star-like if GMG_M is a star for some perfect matching MM, making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.Comment: 21 pages, 7 figure

    Distributed (Δ+1)(\Delta+1)-Coloring in Sublogarithmic Rounds

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    We give a new randomized distributed algorithm for (Δ+1)(\Delta+1)-coloring in the LOCAL model, running in O(log⁡Δ)+2O(log⁥log⁥n)O(\sqrt{\log \Delta})+ 2^{O(\sqrt{\log \log n})} rounds in a graph of maximum degree~Δ\Delta. This implies that the (Δ+1)(\Delta+1)-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of Ω(min⁥(log⁥nlog⁥log⁥n,log⁡Δlog⁥log⁡Δ))\Omega \left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log \Delta}{\log \log \Delta} \right) \right) by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains Δ+1\Delta+1 colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts

    Listing k-cliques in Sparse Real-World Graphs

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    International audienceMotivated by recent studies in the data mining community which require to efficiently list all k-cliques, we revisit the iconic algorithm of Chiba and Nishizeki and develop the most efficient parallel algorithm for such a problem. Our theoretical analysis provides the best asymptotic upper bound on the running time of our algorithm for the case when the input graph is sparse. Our experimental evaluation on large real-world graphs shows that our parallel algorithm is faster than state-of-the-art algorithms, while boasting an excellent degree of parallelism. In particular, we are able to list all k-cliques (for any k) in graphs containing up to tens of millions of edges as well as all 10-cliques in graphs containing billions of edges, within a few minutes and a few hours respectively. Finally, we show how our algorithm can be employed as an effective subroutine for finding the k-clique core decomposition and an approximate k-clique densest subgraphs in very large real-world graphs

    Listing Subgraphs by Cartesian Decomposition

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    We investigate a decomposition technique for listing problems in graphs and set systems. It is based on the Cartesian product of some iterators, which list the solutions of simpler problems. Our ideas applies to several problems, and we illustrate one of them in depth, namely, listing all minimum spanning trees of a weighted graph G. Here iterators over the spanning trees for unweighted graphs can be obtained by a suitable modification of the listing algorithm by [Shioura et al., SICOMP 1997], and the decomposition of G is obtained by suitably partitioning its edges according to their weights. By combining these iterators in a Cartesian product scheme that employs Gray coding, we give the first algorithm which lists all minimum spanning trees of G in constant delay, where the delay is the time elapsed between any two consecutive outputs. Our solution requires polynomial preprocessing time and uses polynomial space
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