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arXiv · 1810.07023

Some local--global phenomena in locally finite graphs

Armen S. Asratian·Jonas B. Granholm·Nikolay K. Khachatryan
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Abstract

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a graph $G$ the ball of radius $r$ centered at $w$ is the subgraph of $G$ induced by the set $M_r(w)$ of vertices whose distance from $w$ does not exceed $r$). In particular, we prove that if every ball of radius 2 in $G$ is 2-connected and $G$ satisfies the condition $d_G(u)+d_G(v)\geq |M_2(w)|-1$ for each path $uwv$ in $G$, where $u$ and $v$ are non-adjacent vertices, then $G$ has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in $G$ satisfies Ore's condition (1960) then all balls of any radius in $G$ are Hamiltonian.

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Armen S. Asratian, Jonas B. Granholm, Nikolay K. Khachatryan. 2018-10-16. Some local--global phenomena in locally finite graphs. https://doi.org/10.1016/j.dam.2019.12.006

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