The k-walks in 2K2-free graphs

• Main Author: Gao, Mou
• Other Authors: Dmitrii V Pasechnik
• Format: Theses and Dissertations
• Language: English
• Published: 2016
• Subjects: DRNTU::Science
• Online Access: http://hdl.handle.net/10356/66352
• Institution: Nanyang Technological University

After a review of Hamiltonicity of graphs and related concepts, we discuss several generalizations of Hamilton cycles: k-walks, k-trees, Hamilton-prisms and edge-dominating cycles, and investigate the relationship between them. In particular, we focus on the Jackson-Wormald conjecture and show that it holds for a graph with an edge-dominating cycle. The latter gives us our central result: an efficient algorithmic proof of Jackson-Wormald conjecture for 2K_2-free graphs. Another main result is that each (1+\epsilon) -tough 2K_2-free graph is prism-Hamiltonian. Generally, being prism-Hamiltonian is a stronger property than admitting k-walks for all k\ge2, but weaker than being traceable. Finally, we present several results on the existence of 2-walks under the 1-toughness assumption for some other graphs, and pose conjectures for further research.