Nowhere-zero flows and colorings of graphs

Author(s)

Candida Nunes da Silva

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Computação
• Defense date: 2009-04-12

Examining board members

Cláudio Leonardo Lucchesi; Daniel Harven Younger; Marcelo Henriques de Carvalho; Arnaldo Mandel; Paulo Feofiloff; Orlando Lee

• Advisor: Cláudio Leonardo Lucchesi

Abstract

The theme of this thesis is nowhere-zero flows and colourings of graphs, two subjects that are closely related. We focus mainly on the three Conjectures of Tutte concerning 5-, 4- and 3-nowhere-zero flows. These conjectures were proposed, respectively, in the 50's, 60's and 70's; all of them remain open so far. In this thesis we present three different approaches for the study of Tutte's Conjectures, with emphasis on the 3-Flow Conjecture. In the first approach, we introduce the concept of flow-critical graph. A graph is flowcritical if it does not admit a nowhere-zero k-flow but, when a simple reduction operation is applied, the resulting graph does admit a nowhere-zero k-flow. The motivation for the study of such graphs is due to the observation that any minimum counterexample for any of Tutte's conjectures lies in this particular class. In the second approach, we study the cyclic-connectivity of a minimum counterexample for an equivalent version of the 3-Flow Conjecture. In the third approach, we give a new proof for Gr¨otzsch's Theorem that differs from the original in the fact that it does not depend on Euler's Formula. (AU)