2,417 research outputs found

    Even-cycle decompositions of graphs with no odd-K4K_4-minor

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    An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no K5K_5-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-K4K_4-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-K4K_4-minor-free' cannot be replaced with `odd-K5K_5-minor-free.' The main technical ingredient is a structural characterization of the class of odd-K4K_4-minor-free graphs, which is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio

    Twisty itsy bitsy topological field theory

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    We extend the topological field theory (``itsy bitsy topological field theory"') of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (``itsy'') and quantum-field-theoretic (``bitsy'') aspects. In the process we extend some results of sutured Floer homology, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.Comment: 52 pages, 26 figure

    Model counting for CNF formuals of bounded module treewidth.

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    The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity

    The inertia of weighted unicyclic graphs

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    Let GwG_w be a weighted graph. The \textit{inertia} of GwG_w is the triple In(Gw)=(i+(Gw),i−(Gw),In(G_w)=\big(i_+(G_w),i_-(G_w), i0(Gw)) i_0(G_w)\big), where i+(Gw),i−(Gw),i0(Gw)i_+(G_w),i_-(G_w),i_0(G_w) are the number of the positive, negative and zero eigenvalues of the adjacency matrix A(Gw)A(G_w) of GwG_w including their multiplicities, respectively. i+(Gw)i_+(G_w), i−(Gw)i_-(G_w) is called the \textit{positive, negative index of inertia} of GwG_w, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order nn with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order nn with two positive, two negative and at least n−6n-6 zero eigenvalues, respectively.Comment: 23 pages, 8figure

    Embedded graph invariants in Chern-Simons theory

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    Chern-Simons gauge theory, since its inception as a topological quantum field theory, has proved to be a rich source of understanding for knot invariants. In this work the theory is used to explore the definition of the expectation value of a network of Wilson lines - an embedded graph invariant. Using a slight generalization of the variational method, lowest-order results for invariants for arbitrary valence graphs are derived; gauge invariant operators are introduced; and some higher order results are found. The method used here provides a Vassiliev-type definition of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not frame individual vertices. Though, without a global projection of the graph, there is an ambiguity in the relation of the decomposition of distinct vertices. It is suggested that framing may be seen as arising from this ambiguity - as a way of relating frames at distinct vertices.Comment: 20 pages; RevTex; with approx 50 ps figures; References added, introduction rewritten, version to be published in Nuc. Phys.
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