Semester

Spring

Date of Graduation

2002

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Hong-Jian Lai.

Abstract

Bondy conjectured that if G is a simple 2-connected graph with n ≥ 3 vertices, then the edges of G can be covered by at most 2n-33 cycles. In Chapter 2, a result on small cycle cover is obtained and we also show that the result is as best as possible.;Thomassen conjectured that every 4-connected line graph is hamiltonian. In Chapters 3 and 4, we apply Catlin's reduction method to study cycles in line graphs. Results about hamiltonian connectivity of line graphs and 3-edge-connected graphs are obtained. Several former results are extended.;Jaeger, Linial, Payan and Tarsi introduced group coloring in 1992 and proved that the group chromatic number for every planar graph is at most 6. It is shown that the bound 6 can be decreased to 5. Jaeger, Linial, Payan and Tarsi also proved that the group chromatic number for every planar graph with girth at least 4 is at most 4. Chapters 5 and 6 are devoted to the study of group coloring of graphs.;The concept of list coloring, choosability and choice number was introduced by Erdos, Rubin and Taylor in 1979 and Vizing in 1976. Alon and Tarsi proved that every bipartite planar graph is 3-choosable. Thomassen showed that every planar graph is 5-choosable and that every planar graph with girth at least 5 is 3-choosable. In Chapter 7, results on list coloring are obtained, extending a former result of Thomassen.

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