Nowhere-Zero 5-Flows On Cubic Graphs with Oddness 4

Tutte’s 5-flow conjecture from 1954 states that every bridge- less graph has a nowhere-zero 5-flow. It suffices to prove the conjecture for cyclically 6-edge-connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow. This implies that every minimum counterexample to the 5-flow conjecture has oddness at least 6

Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic graphs with $\omega \geq 2$. We show that if a cubic graph $G$ has no edge cut with fewer than ${5/2} \omega - 1$ edges that separates two odd cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq {5/2} \omega - 1$, then $G$ has a nowhere-zero 5-flow

Tutte's dichromate for signed graphs

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the ``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.Comment: 53 pp. 9 figure

Algebraic Methods for Reducibility in Nowhere-Zero Flows

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold for all graphs. We then show how to use these algebraic expressions to prove reducibility. In our case, these inequalities and equalities are linear. We can thus use the well developed theory of linear programming to obtain certificates of these proof. We make publicly available computer programs we wrote to generate the algebraic expressions and obtain the certificates

Vertex Magic Group Edge Labelings

A project submitted to the faculty of the graduate school of the University of Minnesota in partial fulfillment of the requirements for the degree of Master of Science. May 2017. Major: Mathematics and Statistics. Advisor: Dalibor Froncek. 1 computer file (PDF); vi, 46 pages, appendix A, Ill. (some col.)A vertex-magic group edge labeling of a graph G(V;E) with |E| = n is an injection from E to an abelian group ᴦ of order n such that the sum of labels of all incident edges of every vertex x ϵ V is equal to the same element µ ϵ ᴦ. We completely characterize all Cartesian products Cn□Cm that admit a vertex-magic group edge labeling by Z2nm, as well as provide labelings by a few other finite abelian groups