Semester

Spring

Date of Graduation

2013

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Cun-Quan Zhang

Committee Co-Chair

Mark Culp

Committee Member

John Goldwasser

Committee Member

Hong-Jian Lai

Committee Member

Jerzy Wojciechowski

Abstract

We primarily consider the problem of finding a family of circuits to cover a bidgeless graph (mainly on cubic graph) with respect to a given weight function defined on the edge set. The first chapter of this thesis is going to cover all basic concepts and notations will be used and a survey of this topic.;In Chapter two, we shall pay our attention to the Strong Circuit Double Cover Conjecture (SCDC Conjecture). This conjecture was verified for some graphs with special structure. As the complement of two factor in cubic graph, the Berge-Fulkersen Conjecture was introduced right after SCDC Conjecture. In Chapter three, we shall present a series of conjectures related to perfect matching covering and point out their relationship.;In last chapter, we shall introduce the saturation number, in contrast to extremal number (or known as Turan Number), and describe the edge spectrum of saturation number for small paths, where the spectrum was consisted of all possible integers between saturation number and Turan number.

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