A generalization of a result of Häggkvist and Nicoghossian

Using a variation of the Bondy-Chvátal closure theorem the following result is proved: If G is a 2-connected graph with n vertices and connectivity κ such that d(x) + d(y) + d(z) ≥ n + κ for any triple of independent vertices x, y, z, then G is hamiltonian

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

A look at cycles containing specified elements of a graph

AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration