Some local–global phenomena in locally finite graphs

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 Mr(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 dG(u) + dG(v) ≥ |M2(w)| − 1 for each path uwv in G, where u and v are non-adjacent vertices, then G has a Hamiltonian curve, introduced by Kündgen, 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. Finally, we show that the k-connectedness of all balls of radius r in a graph G implies the k-connectedness of all balls in G with radius bigger than r. This is a generalization of a result of Chartrand and Pippert (1974).

On Hamiltonicity of regular graphs with bounded second neighborhoods

“…It is easy to verify that G ′ ∈ G (6). It is also clear that G ′ can be constructed from G in polynomial time.…”

“…In this paper we continue our investigation of interconnections between local properties of a graph and its Hamiltonicity (see, for example, [2][3][4][5][6][7][8][9]).…”

“…Clearly, G is locally connected if and only if every ball of radius 1 in G is 2-connected. Some Hamiltonian properties of a graph have been obtained using the structure of balls of radius 2 (see, for example, [3][4][5][6][7][8][9]25,32]). In particular, Asratian [3] obtained the following result:…”