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 $d_G(u)+d_G(v)\geq |M_2(w)|-1$ 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.Comment: 18 pages, 6 figures; journal accepted versio

Some panconnected and pancyclic properties of graphs with a local ore-type condition

Asratian and Khachatrian proved that a connected graphG of order at least 3 is hamiltonian ifd(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| for any pathuwv withuv ∉ E(G), whereN(x) is the neighborhood of a vertexx. We prove that a graphG with this condition, which is not complete bipartite, has the following properties: a) For each pair of verticesx, y with distanced(x, y) ≥ 3 and for each integern, d(x, y) ≤ n ≤ |V(G)| − 1, there is anx − y path of lengthn.  (b)For each edgee which does not lie on a triangle and for eachn, 4 ≤ n ≤ |V(G)|, there is a cycle of lengthn containinge.  (c)Each vertex ofG lies on a cycle of every length from 4 to |V(G)|.  This implies thatG is vertex pancyclic if and only if each vertex ofG lies on a triangle