71 research outputs found

    Decomposition theorem on matchable distributive lattices

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    A distributive lattice structure M(G){\mathbf M}(G) has been established on the set of perfect matchings of a plane bipartite graph GG. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph GG is elementary, then M(G){\mathbf M}(G) is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice L\mathbf{L} is an MDL if and only if each factor in any cartesian product decomposition of L\mathbf{L} is an MDL. Two types of MDLs are presented: J(m×n)J(\mathbf{m}\times \mathbf{n}) and J(T)J(\mathbf{T}), where m×n\mathbf{m}\times \mathbf{n} denotes the cartesian product between mm-element chain and nn-element chain, and T\mathbf{T} is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

    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

    On Disjoint hypercubes in Fibonacci cubes

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    The {\em Fibonacci cube} of dimension nn, denoted as Γ_n\Gamma\_n, is the subgraph of nn-cube Q_nQ\_n induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in Γ_n\Gamma\_n isomorphic to Q_kQ\_k, and denote this number by q_k(n)q\_k(n). We prove several recursive results for q_k(n)q\_k(n), in particular we prove that q_k(n)=q_k1(n2)+q_k(n3)q\_{k}(n) = q\_{k-1}(n-2) + q\_{k}(n-3). We also prove a closed formula in which q_k(n)q\_k(n) is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence {q_k(n)}_n=0\{q\_{k}(n)\}\_{n=0}^{ \infty}

    Outerplane bipartite graphs with isomorphic resonance graphs

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    We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Moreover, two additional characterizations are expressed in terms of resonance digraphs and via local structures of inner duals of 2-connected outerplane bipartite graphs, respectively

    Fibonacci (p, r)-cubes which are median graphs

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    AbstractThe Fibonacci (p, r)-cube is an interconnection topology, which unifies a wide range of connection topologies, such as hypercube, Fibonacci cube, postal network, etc. It is known that the Fibonacci cubes are median graphs [S. Klavžar, On median nature and enumerative properties of Fibonacci-like cubes, Discrete Math. 299 (2005) 145–153]. The question for determining which Fibonacci (p, r)-cubes are median graphs is solved completely in this paper. We show that Fibonacci (p, r)-cubes are median graphs if and only if either r≤p and r≤2, or p=1 and r=n
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