Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph G M is hyperbolic and that δ(G M ) is comparable to diam(G M ). Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs; in fact, it is shown that 5/4 ≤ δ(G M ) ≤ 5/2. Graphs G whose Mycielskian have hyperbolicity constant 5/4 or 5/2 are characterized. The hyperbolicity constants of the Mycielskian of path, cycle, complete and complete bipartite graphs are calculated explicitly. Finally, information on δ(G) just in terms of δ(G M ) is obtained.

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“…(see, for example, [6][7][8][9]14,[18][19][20][21][25][26][27][28][29][30][31][32][33][34][35]). x, y ∈ J(G) \ V(G), z ∈ J(G), …”

Section: Introductionmentioning

confidence: 99%

Gromov Hyperbolicity in Mycielskian Graphs

et al. 2017

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“…In [5], the equivalence of the hyperbolicity of graphs and negatively curved surfaces was proved. The study of hyperbolic graphs is a topic of increasing interest (see, e.g., and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Mathematical Properties on the Hyperbolicity of Interval Graphs

et al. 2017

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Abstract:Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.

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“…The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

“…In [25,Section 1.3] is observed that the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it (see also [14,16,17]). Hence, establishing hyperbolicity criteria for graphs will be of interest to us.…”

Section: Introductionmentioning

confidence: 99%