This article is intended as a survey, updating earlier surveys in the area. For completeness
of the presentation of both particular questions and the general area, it also contains …

We survey results and open problems in hamiltonian graph theory centered around two
conjectures of the 1980s that are still open: every 4-connected claw-free graph (line graph) …

A graph G is Hamilton-connected if for any pair of distinct vertices u, v ∈ V (G) u, v∈ V (G), G
has a spanning (u, v)-path; G is 1-hamiltonian if for any vertex subset S ⊆ V (G) S⊆ V (G) …

A graph G is said to be 1-tough if for every vertex cut S of G, the number of components of
G− S does not exceed| S|. Being 1-tough is an obvious necessary condition for a graph to be …

We describe a new type of sufficient condition for a balanced bipartite digraph to be
hamiltonian. Let D be a balanced bipartite digraph and x,y be distinct vertices in D. {x,y\} …

The concept of toughness was introduced by Chvátal [34] more than forty years ago.
Toughness resembles vertex connectivity, but is different in the sense that it takes into …

In 1972, Woodall raised the following Ore-type condition for directed Hamilton cycles in
digraphs: Let D be a digraph. If for every vertex pair u and v, where there is no arc from u to …

Abstract For integers s 1, s 2, s 3> 0, let N s 1, s 2, s 3 denote the graph obtained by
identifying each vertex of a K 3 with an end vertex of three disjoint paths P s 1+ 1, P s 2+ 1, P …

A grid graph is a finite embedded subgraph of the infinite integer grid. A solid grid graph is a
grid graph without holes, ie, each bounded face of the graph is a unit square. The …

A cycle C in a graph G is called dominating if every edge of G is incident with a vertex of C.
For a set H of connected graphs, a graph G is said to be H-free if G does not contain any …