Algebraic Approaches to Flow and Cycle Conjectures for Graphs

Louisiana State University, Baton Rouge LA

Investigators

Abstract

Algebraic Approaches to Flow and Cycle Conjectures for Graphs. Abstract: This proposed project will focus on applying certain algebraic ideas in an attempt to prove the major open conjectures about flows and cycle covers in graphs. The ultimate goal is to prove all of the major open conjectures about flows and cycle covers in graphs, while the realistic expectation is to produce plenty of good mathematics in the attempt. Two big conjectures in this area are Tutte's 5-flow conjecture (T5FC) and the Cycle Double Cover Conjecture (CDCC) of Szekeres and various strengthenings of CDCC, including the Orientable Five Cycle Double Cover Conjecture (O5CDCC) of Archdeacon and Jaeger, which implies both T5FC and CDCC and many other strengthenings of CDCC. These in turn are all implied by the PI's Petersen-or-K_4 Flow Conjecture (P10vK4FC), the proof of which is the ultimate goal of the proposed project. These conjectures, which are very well known to graph theorists, especially T5FC and CDCC, are in the same family of mathematical problems as the Four Color Theorem (4CT) which is even much more widely known to mathematicians and the general public. Most research on these problems depends primarily on combinatorial graph theoretic techniques. The novelty of the PI's approach is to apply many algebraic, as well as topological, ideas in addition to all the usual combinatorial graph theoretic ones.