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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).

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Let F be a (possibly improper) edge-coloring of a graph G; a vertex coloring of G is adapted to F if no color appears at the same time on an edge and on its two endpoints. If for some integer k, a graph G is such that given any list assignment L to the vertices of G, with |L(v)| ≥ k for all v, and any edge-coloring F of G, G admits a coloring c adapted to F where c(v) ∈ L(v) for all v, then G is said to be adaptably k-choosable. A (k, d)-list assignment for a graph G is a map that assigns to each vertex v a list L(v) of at least k colors such that |L(x) ∩ L(y)| ≤ d whenever x and y are adjacent. A graph is (k, d)-choosable if for every (k, d)-list assignment L there is an L-coloring of G. It has been conjectured that planar graphs are (3, 1)-choosable. We give some progress on this conjecture by giving sufficient conditions for a planar graph to be adaptably 3-choosable. Since (k, 1)-choosability is a special case of adaptable k-choosablity, this implies that a planar graph satisfying these conditions is (3, 1)-choosable.

Let G(k) denote the set of connected k-regular graphs G, k ≥ 2, where the number of vertices at distance 2 from any vertex in G does not exceed k. Asratian ( 2006) showed (using other terminology) that a graph G ∈ G(k) is Hamiltonian if for each vertex u of G the subgraph induced by the set of vertices at distance at most 2 from u is 2-connected. We prove here that in fact all graphs in the sets G(3), G(4) and G(5) are Hamiltonian. We also prove that the problem of determining whether there exists a Hamilton cycle in a graph from G( 6) is NP-complete. Nevertheless we show that every locally connected graph G ∈ G(k), k ≥ 6, is Hamiltonian and that for each non-Hamiltonian cycle C in G there exists a cycleFinally, we note that all our conditions for Hamiltonicity apply to infinitely many graphs with large diameters.

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for every triple of vertices u, v, w where u and v are at distance 2 and w ∈ N (u) ∩ N (v). proved that all finite connected L1-graphs on at least three vertices such that |N (u) ∩ N (v)| ≥ 2 for each pair of vertices u, v at distance 2 are Hamiltonian, except for a simple family K of exceptions.We show that not all such graphs are pancyclic, but that any non-Hamiltonian cycle in such a graph can be extended to a larger cycle containing all vertices of the original cycle and at most two other vertices. We also prove a similar result for paths whose endpoints do not have any common neighbors.