Ronald J. Gould

Advances on the Hamiltonian Problem – A Survey

Abstract. 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 material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs and digraphs.

Updating the Hamiltonian problem—a survey

This is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle. This article also contains some material on related topics such as traceable, hamiltonian-connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area.

Characterizing forbidden pairs for hamiltonian properties

Abstract In this paper we characterize those pairs of forbidden subgraphs sufficient to imply various hamiltonian type properties in graphs. In particular, we find all forbidden pairs sufficient, along with a minor connectivity condition, to imply a graph is traceable, hamiltonian, pancyclic, panconnected or cycle extendable. We also consider the case of hamiltonian-connected graphs and present a result concerning the pairs for such graphs.

Degree conditions for 2-factors

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ores classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k.

Neighborhood unions and hamiltonian properties in graphs

We investigate the relationship between the cardinality of the union of the neighborhoods of an arbitrary pair of nonadjacent vertices and various hamiltonian type properties in graphs. In particular, we show that if G is 2-connected, of order p ≥ 3 and if for every pair of nonadjacent vertices x and y: 1. (a) ∥N(x) ⌣ N(y)∥ ≧ (p − 1)2, then G is traceable, 2. (b) ∥N(x) ⌣ N(y)∥ ≧ (2p − 1)3, then G is hamiltonian, and if G is 3-connected and 3. (c) ∥N(x) ⌣ N(y)∥ ≧ 2p3, then G is hamiltonian-connected.

Forbidden subgraphs and Hamiltonian properties of graphs

Various sufficient conditions are given, in terms of forbidden subgraphs, that imply a graph is either homogeneously traceable, hamiltonian or pancyclic.

On k-ordered graphs

We prove a hypergraph version of Halls theorem. The proof is topological.

Recent Advances on the Hamiltonian Problem: Survey III

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 some material on closely related topics such as traceable, pancyclic and Hamiltonian connected graphs.

Extremal graphs for intersecting triangles

Abstract It is known that for a graph on n vertices [ n 2 /4] + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex.

Partitioning Vertices of a Tournament into Independent Cycles

Let k be a positive integer. A strong digraph G is termed k-connected if the removal of any set of fewer than k vertices results in a strongly connected digraph. The purpose of this paper is to show that every k-connected tournament with at least 8k vertices contains k vertex-disjoint directed cycles spanning the vertex set. This result answers a question posed by Bollobas.