Notes on sufficient conditions for a graph to be Hamiltonian

Michael Joseph Paul,Carmen Baytan Shershin,Anthony Connors Shershin

doi:10.1155/s0161171291001138

Abstract

The first part of this paper deals with an extension of Dirac′s Theorem to directed graphs. It is related to a result often referred to as the Ghouila‐Houri Theorem. Here we show that the requirement of being strongly connected in the hypothesis of the Ghouila‐Houri Theorem is redundant.The Second part of the paper shows that a condition on the number of edges for a graph to be hamiltonian implies Ore′s condition on the degrees of the vertices.

Highlights

The first part of this paper deals with an extension of Dirac’s Theorem to directed graphs
Sufficient conditions for a graph, or digraph, to have a hamilton cycle usually take the form of implicitly requiring many edges
The purpose of this section is to show that the edge condition implies Ore’s condition, to give simple examples to show that no implication exists between Dirac’s condition and the edge condition, and to show that Ore’s condition does not necessarily imply either Dirac’s or the edge conditions: The set diagram below (Figure 1) summarizes the relationships among hamilton graphs satisfying Ore’s (0), Dirac’s (D), and/or the edge condition (E)

Summary

Introduction

The first part of this paper deals with an extension of Dirac’s Theorem to directed graphs. Hamilton cycle, directed graphs, Ghouila-Houri Theorem, strongly connected digraphs, Dirac’s Theorem, Ore’s Theorem, edge condition. Sufficient conditions for a graph, or digraph, to have a hamilton cycle usually take the form of implicitly requiring many edges.

Results

Conclusion