Displaying similar documents to “The hyperbolicity constant of infinite circulant graphs”

Gromov hyperbolic cubic graphs

Domingo Pestana, José Rodríguez, José Sigarreta, María Villeta (2012)

Open Mathematics

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If X is a geodesic metric space and x 1; x 2; x 3 ∈ X, a geodesic triangle T = {x 1; x 2; x 3} is the union of the three geodesics [x 1 x 2], [x 2 x 3] and [x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity...

Gromov hyperbolicity of planar graphs

Alicia Cantón, Ana Granados, Domingo Pestana, José Rodríguez (2013)

Open Mathematics

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We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ2 such that every tile is a triangle and a partial answer to this question is given....

Products of Geodesic Graphs and the Geodetic Number of Products

Jake A. Soloff, Rommy A. Márquez, Louis M. Friedler (2015)

Discussiones Mathematicae Graph Theory

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Given a connected graph and a vertex x ∈ V (G), the geodesic graph Px(G) has the same vertex set as G with edges uv iff either v is on an x − u geodesic path or u is on an x − v geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of Km,n with itself, where m, n ≥ 4, is equal to the minimum of m, n and eight.

Non-hyperbolicity in random regular graphs and their traffic characteristics

Gabriel Tucci (2013)

Open Mathematics

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In this paper we prove that random d-regular graphs with d ≥ 3 have traffic congestion of the order O(n logd−13 n) where n is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically δ-hyperbolic for any non-negative δ almost surely as n → ∞.

Structure of geodesics in the Cayley graph of infinite Coxeter groups

Ryszard Szwarc (2003)

Colloquium Mathematicae

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Let (W,S) be a Coxeter system such that no two generators in S commute. Assume that the Cayley graph of (W,S) does not contain adjacent hexagons. Then for any two vertices x and y in the Cayley graph of W and any number k ≤ d = dist(x,y) there are at most two vertices z such that dist(x,z) = k and dist(z,y) = d - k. Allowing adjacent hexagons, but assuming that no three hexagons can be adjacent to each other, we show that the number of such intermediate vertices at a given distance from...

Union of Distance Magic Graphs

Sylwia Cichacz, Mateusz Nikodem (2017)

Discussiones Mathematicae Graph Theory

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A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ from V to the set {1, . . . , n} such that the weight w(x) = ∑y∈NG(x) ℓ(y) of every vertex x ∈ V is equal to the same element μ, called the magic constant. In this paper, we study unions of distance magic graphs as well as some properties of such graphs.

Radio Graceful Hamming Graphs

Amanda Niedzialomski (2016)

Discussiones Mathematicae Graph Theory

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For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G) → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u) − f(v)| ≥ k + 1 − d(u, v). We consider k-radio labelings of G when k = diam(G). In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G)|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio...

1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs

Sebastian Hensel, Piotr Przytycki, Richard C. H. Webb (2015)

Journal of the European Mathematical Society

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We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

Orientation distance graphs revisited

Wayne Goddard, Kiran Kanakadandi (2007)

Discussiones Mathematicae Graph Theory

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The orientation distance graph 𝓓ₒ(G) of a graph G is defined as the graph whose vertex set is the pair-wise non-isomorphic orientations of G, and two orientations are adjacent iff the reversal of one edge in one orientation produces the other. Orientation distance graphs was introduced by Chartrand et al. in 2001. We provide new results about orientation distance graphs and simpler proofs to existing results, especially with regards to the bipartiteness of orientation distance graphs...

A note on uniquely H-colourable graphs

Anthony Bonato (2007)

Discussiones Mathematicae Graph Theory

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For a graph H, we compare two notions of uniquely H-colourable graphs, where one is defined via automorphisms, the second by vertex partitions. We prove that the two notions of uniquely H-colourable are not identical for all H, and we give a condition for when they are identical. The condition is related to the first homomorphism theorem from algebra.

On dually compact closed classes of graphs and BFS-constructible graphs

Norbert Polat (2003)

Discussiones Mathematicae Graph Theory

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A class C of graphs is said to be dually compact closed if, for every infinite G ∈ C, each finite subgraph of G is contained in a finite induced subgraph of G which belongs to C. The class of trees and more generally the one of chordal graphs are dually compact closed. One of the main part of this paper is to settle a question of Hahn, Sands, Sauer and Woodrow by showing that the class of bridged graphs is dually compact closed. To prove this result we use the concept of constructible...

Light Graphs In Planar Graphs Of Large Girth

Peter Hudák, Mária Maceková, Tomáš Madaras, Pavol Široczki (2016)

Discussiones Mathematicae Graph Theory

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A graph H is defined to be light in a graph family 𝒢 if there exist finite numbers φ(H, 𝒢) and w(H, 𝒢) such that each G ∈ 𝒢 which contains H as a subgraph, also contains its isomorphic copy K with ΔG(K) ≤ φ(H, 𝒢) and ∑x∈V(K) degG(x) ≤ w(H, 𝒢). In this paper, we investigate light graphs in families of plane graphs of minimum degree 2 with prescribed girth and no adjacent 2-vertices, specifying several necessary conditions for their lightness and providing sharp bounds on φ and w...