#### Semester

Fall

#### Date of Graduation

2001

#### Document Type

Dissertation

#### Degree Type

PhD

#### College

Eberly College of Arts and Sciences

#### Department

Mathematics

#### Committee Chair

Cun-Quan Zhang.

#### Abstract

This thesis is on a long standing open conjecture proposed by one of the most prominent mathematicians, Dr. C. Thomassen: Every longest circuit of 3-connected graph has a chord. In 1987, C. Q. Zhang proved that every longest circuit of a 3-connected planar graph G has a chord if G is cubic or if the minimum degree is at least 4. In 1997, Carsten Thomassen proved that every longest circuit of 3-connected cubic graph has a chord.;In this dissertation, we prove the following three independent partial results: (1) Every longest circuit of a 3-connected graph embedded in a projective plane with minimum degree at least has a chord (Theorem 2.3.1). (2) Every longest circuit of a 3-connected cubic graph has at least two chords. Furthermore if the graph is also a planar, then every longest circuit has at least three chords (Theorem 3.2.6, 3.2.7). (3) Every longest circuit of a 4-connected graph embedded in a torus or Klein bottle has a chord.;We get these three independent results with three totally different approaches: Connectivity (Tutte circuit), second Hamilton circuit, and charge and discharge methods.

#### Recommended Citation

Li, Xuechao, "Chords of longest circuits of graphs" (2001). *Graduate Theses, Dissertations, and Problem Reports*. 1450.

https://researchrepository.wvu.edu/etd/1450