266 research outputs found

    Covering line graphs with equivalence relations

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    An equivalence graph is a disjoint union of cliques, and the equivalence number eq(G)\mathit{eq}(G) of a graph GG is the minimum number of equivalence subgraphs needed to cover the edges of GG. We consider the equivalence number of a line graph, giving improved upper and lower bounds: 13log2log2χ(G)<eq(L(G))2log2log2χ(G)+2\frac 13 \log_2\log_2 \chi(G) < \mathit{eq}(L(G)) \leq 2\log_2\log_2 \chi(G) + 2. This disproves a recent conjecture that eq(L(G))\mathit{eq}(L(G)) is at most three for triangle-free GG; indeed it can be arbitrarily large. To bound eq(L(G))\mathit{eq}(L(G)) we bound the closely-related invariant σ(G)\sigma(G), which is the minimum number of orientations of GG such that for any two edges e,fe,f incident to some vertex vv, both ee and ff are oriented out of vv in some orientation. When GG is triangle-free, σ(G)=eq(L(G))\sigma(G)=\mathit{eq}(L(G)). We prove that even when GG is triangle-free, it is NP-complete to decide whether or not σ(G)3\sigma(G)\leq 3.Comment: 10 pages, submitted in July 200

    Exponential Time Complexity of the Permanent and the Tutte Polynomial

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    We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting

    Bounded degree graphs and hypergraphs with no full rainbow matchings

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    Given a multi-hypergraph GG that is edge-colored into color classes E1,,EnE_1, \ldots, E_n, a full rainbow matching is a matching of GG that contains exactly one edge from each color class EiE_i. One way to guarantee the existence of a full rainbow matching is to have the size of each color class EiE_i be sufficiently large compared to the maximum degree of GG. In this paper, we apply a simple iterative method to construct edge-colored multi-hypergraphs with a given maximum degree, large color classes, and no full rainbow matchings. First, for every r1r \ge 1 and Δ2\Delta \ge 2, we construct edge-colored rr-uniform multi-hypergraphs with maximum degree Δ\Delta such that each color class has size EirΔ1|E_i| \ge r\Delta - 1 and there is no full rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and Meshulam (2005) is best possible. Second, we construct properly edge-colored multigraphs with no full rainbow matchings which disprove conjectures of Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings to list edge-colorings and prove that a color degree generalization of Galvin's theorem (1995) does not hold

    Strong edge-colorings for k-degenerate graphs

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    We prove that the strong chromatic index for each kk-degenerate graph with maximum degree Δ\Delta is at most (4k2)Δk(2k1)+1(4k-2)\Delta-k(2k-1)+1
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