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Symmetry 2017, 9(8), 131; doi:10.3390/sym9080131

Gromov Hyperbolicity in Mycielskian Graphs

1
Department of Mathematics and Computer Science, Saint Louis University, Avenida del Valle 34, 28003 Madrid, Spain
2
Department of Mathematics, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
*
Author to whom correspondence should be addressed.
Academic Editor: Angel Garrido
Received: 21 June 2017 / Revised: 14 July 2017 / Accepted: 21 July 2017 / Published: 27 July 2017
(This article belongs to the Special Issue Graph Theory)
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Abstract

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. View Full-Text
Keywords: extremal problems on graphs; Mycielskian graphs; geodesics; Gromov hyperbolicity extremal problems on graphs; Mycielskian graphs; geodesics; Gromov hyperbolicity
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MDPI and ACS Style

Granados, A.; Pestana, D.; Portilla, A.; Rodríguez, J.M. Gromov Hyperbolicity in Mycielskian Graphs. Symmetry 2017, 9, 131.

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