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Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

David Cimasoni (2012)

Journal of the European Mathematical Society

Let ∑ be a flat surface of genus g with cone type singularities. Given a bipartite graph Γ isoradially embedded in ∑ , we define discrete analogs of the 2 2 g Dirac operators on ∑ . These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair Γ ⊂ ∑ for these discrete Dirac operators to be Kasteleyn matrices of the graph Γ . As a consequence, if these conditions are met, the partition function of the dimer...

Disjoint 5-cycles in a graph

Hong Wang (2012)

Discussiones Mathematicae Graph Theory

We prove that if G is a graph of order 5k and the minimum degree of G is at least 3k then G contains k disjoint cycles of length 5.

Disjoint triangles and quadrilaterals in a graph

Hong Wang (2008)

Open Mathematics

Let n, s and t be three integers with s ≥ 1, t ≥ 0 and n = 3s + 4t. Let G be a graph of order n such that the minimum degree of G is at least (n + s)/2. Then G contains a 2-factor with s + t components such that s of them are triangles and t of them are quadrilaterals.

Dissimilarités multivoies et généralisations d'hypergraphes sans triangles

Jean Diatta (1997)

Mathématiques et Sciences Humaines

Les dissimilarités multivoies sont une généralisation naturelle des dissimilarités usuelles deux voies. Dans ce papier, des classes de dissimilarités multivoies sont étudiées, ainsi que des modèles de passage d'un nombre de voies donné à un autre nombre de voies. Une application à la spécification de systèmes classifiants a conduit à une bijection entre une classe de dissimilarités multivoies et une famille de systèmes stratifiés de classifccation.

Distance 2-Domination in Prisms of Graphs

Ferran Hurtado, Mercè Mora, Eduardo Rivera-Campo, Rita Zuazua (2017)

Discussiones Mathematicae Graph Theory

A set of vertices D of a graph G is a distance 2-dominating set of G if the distance between each vertex u ∊ (V (G) − D) and D is at most two. Let γ2(G) denote the size of a smallest distance 2-dominating set of G. For any permutation π of the vertex set of G, the prism of G with respect to π is the graph πG obtained from G and a copy G′ of G by joining u ∊ V(G) with v′ ∊ V(G′) if and only if v′ = π(u). If γ2(πG) = γ2(G) for any permutation π of V(G), then G is called a universal γ2-fixer. In this...

Distance coloring of the hexagonal lattice

Peter Jacko, Stanislav Jendrol' (2005)

Discussiones Mathematicae Graph Theory

Motivated by the frequency assignment problem we study the d-distant coloring of the vertices of an infinite plane hexagonal lattice H. Let d be a positive integer. A d-distant coloring of the lattice H is a coloring of the vertices of H such that each pair of vertices distance at most d apart have different colors. The d-distant chromatic number of H, denoted χ d ( H ) , is the minimum number of colors needed for a d-distant coloring of H. We give the exact value of χ d ( H ) for any d odd and estimations for any...

Distance defined by spanning trees in graphs

Gary Chartrand, Ladislav Nebeský, Ping Zhang (2007)

Discussiones Mathematicae Graph Theory

For a spanning tree T in a nontrivial connected graph G and for vertices u and v in G, there exists a unique u-v path u = u₀, u₁, u₂,..., uₖ = v in T. A u-v T-path in G is a u- v path u = v₀, v₁,...,vₗ = v in G that is a subsequence of the sequence u = u₀, u₁, u₂,..., uₖ = v. A u-v T-path of minimum length is a u-v T-geodesic in G. The T-distance d G | T ( u , v ) from u to v in G is the length of a u-v T-geodesic. Let geo(G) and geo(G|T) be the set of geodesics and the set of T-geodesics respectively in G. Necessary...

Distance in graphs

Roger C. Entringer, Douglas E. Jackson, D. A. Snyder (1976)

Czechoslovak Mathematical Journal

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