26,473 research outputs found

    Resonance graphs of plane bipartite graphs as daisy cubes

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    We characterize all plane bipartite graphs whose resonance graphs are daisy cubes and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if GG is a plane elementary bipartite graph other than K2K_2, then the resonance graph R(G)R(G) is a daisy cube if and only if the Fries number of GG equals the number of finite faces of GG, which in turn is equivalent to GG being homeomorphically peripheral color alternating. Next, we extend the above characterization from plane elementary bipartite graphs to all plane bipartite graphs and show that the resonance graph of a plane bipartite graph GG is a daisy cube if and only if GG is weakly elementary bipartite and every elementary component of GG other than K2K_2 is homeomorphically peripheral color alternating. Along the way, we prove that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes

    Bose-Hubbard model on two-dimensional line graphs

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    We construct a basis for the many-particle ground states of the positive hopping Bose-Hubbard model on line graphs of finite 2-connected planar bipartite graphs at sufficiently low filling factors. The particles in these states are localized on non-intersecting vertex-disjoint cycles of the line graph which correspond to non-intersecting edge-disjoint cycles of the original graph. The construction works up to a critical filling factor at which the cycles are close-packed.Comment: 9 pages, 5 figures, figures and conclusions update

    A construction for a counterexample to the pseudo 2-factor isomorphic graph conjecture

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    A graph GG admiting a 22-factor is \textit{pseudo 22-factor isomorphic} if the parity of the number of cycles in all its 22-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo 22-factor isomorphic regular bipartite graphs. Journal of Combinatorial Theory, Series B, 98(2) (2008), 432-444.] some of the authors of this note gave a partial characterisation of pseudo 22-factor isomorphic bipartite cubic graphs and conjectured that K3,3K_{3,3}, the Heawood graph and the Pappus graph are the only essentially 44-edge-connected ones. In [J. Goedgebeur. A counterexample to the pseudo 22-factor isomorphic graph conjecture. Discr. Applied Math., 193 (2015), 57-60.] Jan Goedgebeur computationally found a graph G\mathscr{G} on 3030 vertices which is pseudo 22-factor isomorphic cubic and bipartite, essentially 44-edge-connected and cyclically 66-edge-connected, thus refuting the above conjecture. In this note, we describe how such a graph can be constructed from the Heawood graph and the generalised Petersen graph GP(8,3)GP(8,3), which are the Levi graphs of the Fano 737_3 configuration and the M\"obius-Kantor 838_3 configuration, respectively. Such a description of G\mathscr{G} allows us to understand its automorphism group, which has order 144144, using both a geometrical and a graph theoretical approach simultaneously. Moreover we illustrate the uniqueness of this graph

    Bipartite induced density in triangle-free graphs

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    We prove that any triangle-free graph on nn vertices with minimum degree at least dd contains a bipartite induced subgraph of minimum degree at least d2/(2n)d^2/(2n). This is sharp up to a logarithmic factor in nn. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of n/dn/d and (2+o(1))n/logn(2+o(1))\sqrt{n/\log n} as nn\to\infty. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most O(min{n,(nlogn)/d})O(\min\{\sqrt{n},(n\log n)/d\}) as nn\to\infty. Relatedly, we also make two conjectures. First, any triangle-free graph on nn vertices has fractional chromatic number at most (2+o(1))n/logn(\sqrt{2}+o(1))\sqrt{n/\log n} as nn\to\infty. Second, any triangle-free graph on nn vertices has list chromatic number at most O(n/logn)O(\sqrt{n/\log n}) as nn\to\infty.Comment: 20 pages; in v2 added note of concurrent work and one reference; in v3 added more notes of ensuing work and a result towards one of the conjectures (for list colouring

    Faster exponential-time algorithms in graphs of bounded average degree

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    We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and exponential space for a constant \eps_d depending only on d, where the O*-notation suppresses factors polynomial in the input size. Thus, we generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of bounded degree. Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorithm for counting perfect matchings in an n-vertex graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on inclusion-exclusion principle instead of algebraic transformations. Building upon this result, we show that the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the constant obtained by us for the Traveling Salesman Problem in graphs of average degree at most 2d. Moreover we obtain a simple algorithm that counts the number of perfect matchings in an n-vertex bipartite graph of average degree at most d in O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].Comment: 10 page

    On the perfect 1-factorisation problem for circulant graphs of degree 4

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    A 1-factorisation of a graph G is a partition of the edge set of G into 1 factors (perfect matchings); a perfect 1-factorisation of G is a 1-factorisation of G in which the union of any two of the 1-factors is a Hamilton cycle in G. It is known that for bipartite 4-regular circulant graphs, having order 2 (mod 4) is a necessary (but not sufficient) condition for the existence of a perfect 1-factorisation. The only known non-bipartite 4-regular circulant graphs that admit a perfect 1-factorisation are trivial (on 6 vertices). We prove several construction results for perfect 1-factorisations of a large class of bipartite 4-regular circulant graphs. In addition, we show that no member of an infinite family of non-bipartite 4-regular circulant graphs admits a perfect 1-factorisation. This supports the conjecture that there are no perfect 1-factorisations of any connected non-bipartite 4-regular circulant graphs of order at least 8

    Maximum Matching via Maximal Matching Queries

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    We study approximation algorithms for Maximum Matching that are given access to the input graph solely via an edge-query maximal matching oracle. More specifically, in each round, an algorithm queries a set of potential edges and the oracle returns a maximal matching in the subgraph spanned by the query edges that are also contained in the input graph. This model is more general than the vertex-query model introduced by binti Khalil and Konrad [FSTTCS\u2720], where each query consists of a subset of vertices and the oracle returns a maximal matching in the subgraph of the input graph induced by the queried vertices. In this paper, we give tight bounds for deterministic edge-query algorithms for up to three rounds. In more detail: 1) As our main result, we give a deterministic 3-round edge-query algorithm with approximation factor 0.625 on bipartite graphs. This result establishes a separation between the edge-query and the vertex-query models since every deterministic 3-round vertex-query algorithm has an approximation factor of at most 0.6 [binti Khalil, Konrad, FSTTCS\u2720], even on bipartite graphs. Our algorithm can also be implemented in the semi-streaming model of computation in a straightforward manner and improves upon the state-of-the-art 3-pass 0.6111-approximation algorithm by Feldman and Szarf [APPROX\u2722] for bipartite graphs. 2) We show that the aforementioned algorithm is optimal in that every deterministic 3-round edge-query algorithm has an approximation factor of at most 0.625, even on bipartite graphs. 3) Last, we also give optimal bounds for one and two query rounds, where the best approximation factors achievable are 1/2 and 1/2 + ?(1/n), respectively, where n is the number of vertices in the input graph
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