Some local-global phenomena in locally finite graphs

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) >= vertical bar M-2(w )vertical bar - 1 for each path uwv in G, where u and v are non-adjacent vertices, then G has a Hamiltonian curve, introduced by Kundgen et al. (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. (C) 2019 Elsevier B.V. All rights reserved.