Date of Award:

5-2014

Document Type:

Dissertation

Degree Name:

Doctor of Philosophy (PhD)

Department:

Mathematics and Statistics

Committee Chair(s)

David E. Brown

Committee

David E. Brown

Committee

LeRoy B. Beasley

Committee

James Powell

Committee

Luis Gordillo

Committee

Richard Lundgren

Abstract

A significant portion of Graph Theory is devoted to determining the characteristics which guarantee the existence of long cycles.

Long cycles have roles in applications to civil engineering, chemistry, and communications, among many others, but the problem, in and of itself, of determining whether a graph has a cycle of some fixed and typically large length is one of the most important problems of both pure Mathematics and Computer Science.

A cycle containing all the vertices of the graph is called a Hamiltonian cycle, and a graph which possesses such a cycle is said to be Hamiltonian. If a graph contains cycles of every length, from three to the number of vertices of the graph it is said to be pancyclic. J.A. Bondy famously posited what is referred to as Bondy’s metaconjecture: Every condition which guarantees a graph is Hamiltonian actually guarantees it is pancyclic. If the vertices of a cycle of length L are contained in a cycle of length L + 1, the cycle is said to be extendable. If every non-Hamiltonian cycle in a graph is extendable, the graph is said to be cycle-extendable. That a graph is cycle extendable is a stronger structural property than being pancyclic which is in turn stronger than being Hamiltonian. Nevertheless George Hendry pioneered the study of cycle extendability by proving the following statement for many types of graphs possessing conditions sufficient for Hamiltonicity: A non-Hamiltonian cycle in a graph with a property that guarantees it is Hamiltonian also guarantees it is cycle extendable.

A graph H is an induced subgraph of another graph G if H can be obtained from G by deleting vertices. A graph is said to be chordal if it has no cycle on four or more vertices as an induced subgraph; that is, every cycle long enough to have a chord (an edge connecting two nonconsecutive vertices on a cycle), has a chord. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable.

In this thesis the following questions are addressed: What are sufficient conditions for cycle extendability? And what progress can we make in resolving Hendry’s Conjecture, in particular? Included among other things, results relating to Hamiltonicity, pancyclicity, and cycle extendability are developed and proved.

It is proved that a graph satisfying the Chv`atal-Erd˝os condition, κ(G) ≥ α(G), is cycle extendable. It is proved that 2-connected claw-free chordal graphs are cycle extendable, and a forbidden subgraph pair determining cycle extendability in chordal graphs is provided. It is also proved that a graph having a Hamiltonian elimination ordering is cycle extendable.

A new minimum degree condition for Hamiltonicity is presented, and generalizations of Dirac’s condition and Ore’s condition are given.

Furthermore, bipartite graphs are investigated, (bipartite graphs being graphs that contain no odd cycles), and the questions are considered: what conditions guarantee there is a Hamiltonian cycle which avoids certain subsets of edges (referred to as “edgeavoiding” Hamiltonicity in a bipartite graph)? and under what conditions, given a bipartite graph G, and any graph F, can we determine whether G contains a Hamiltonian cycle that avoids some subgraph H that is isomorphic to F (such a graph G will be referred to as F-avoiding Hamiltonian bipartite)?

This dissertation presents new conditions and properties that guarantee desired cycle structure in graphs of different kinds, and ultimately contributes to a deeper understanding of the properties of graphs that affect Hamiltonicity and cycle extendability, which is important in many applications in the real world.

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