Gromov Hyperbolicity in Cartesian Product Graphs

Proceedings Mathematical Sciences

María Villeta¹  José M Sigarreta³  Junior Michel²  José M Rodríguez⁴ 

[1] $$;Departamento de Matemáticas, Universidad Carlos III de Madrid, Av. de la Universidad 0, Leganés, Madrid, Spain$$;Departamento de Estadística e Investigación Operativa III, Universidad Complutense de Madrid, Av. Puerta de Hierro s/n., 00 Madrid, Spain$$;Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame , Col. La Garita, Acapulco, Guerrero, México$$

关键词: Infinite graphs;    Cartesian product graphs;    connectivity;    geodesics;    Gromov hyperbolicity.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

If 𝑋 is a geodesic metric space and $x_1,x_2,x_3in X$, a geodesic triangle $T={x_1,x_2,x_3}$ is the union of the three geodesics $[x_1x_2], [x_2x_3]$ and $[x_3x_1]$ 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. If 𝑋 is hyperbolic, we denote by 𝛿(𝑋) the sharp hyperbolicity constant of 𝑋, i.e. $𝛿(X)=inf{𝛿≥ 0:X, ext{is}𝛿-ext{hyperbolic}}$. In this paper we characterize the product graphs 𝐺1 × 𝐺2 which are hyperbolic, in terms of 𝐺1 and 𝐺2: the product graph 𝐺1 × 𝐺2 is hyperbolic if and only if 𝐺1 is hyperbolic and 𝐺2 is bounded or 𝐺2 is hyperbolic and 𝐺1 is bounded. We also prove some sharp relations between the hyperbolicity constant of $𝐺1 × 𝐺2,𝛿(𝐺1),𝛿(𝐺2) and the diameters of 𝐺1 and 𝐺2 (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the precise value of the hyperbolicity constant for many product graphs.

【 授权许可】

Unknown