Bounded-Excess Flows in Cubic Graphs

TL;DR

This paper introduces a new type of flow called (r,alpha)-flow in cubic graphs, proves the existence of a (3.5,0.5)-flow in all cubic graphs, and explores the structure of flow strength using the trace parameter, advancing towards the 5-flow conjecture.

Contribution

The paper establishes that every cubic graph admits a (3.5,0.5)-flow, providing a significant step towards the 5-flow conjecture and analyzing the flow strength poset structure.

Findings

Every cubic graph admits a (3.5,0.5)-flow.

The trace parameter helps compare flow strengths.

The result is optimal for all cubic graphs.

Abstract

An (r,alpha)-bounded excess flow ((r,alpha)-flow) in an orientation of a graph G=(V,E) is an assignment of a real "flow value" between 1 and r-1 to every edge. Rather than 0 as in an actual flow, some flow excess, which does not exceed alpha may accumulate in any vertex. Bounded excess flows suggest a generalization of Circular nowhere zero flows, which can be regarded as (r,0)-flows. We define (r,alpha) as Stronger or equivalent to (s,beta) If the existence of an (r,alpha)-flow in a cubic graph always implies the existence of an (s,beta)-flow in the same graph. Then we study the structure of the two-dimensional flow strength poset. A major role is played by the "Trace" parameter: tr(r,alpha)=(r-2alpha) divided by (1-alpha). Among points with the same trace the stronger is the one with the larger r (an r-cnzf is of trace r). About one half of the article is devoted to proving the main result: Every cubic graph admits a (3.5,0.5)-flow. tr(3.5,0.5)=5 so it can be considered a step in the direction of the 5-flow Conjecture. Our result is best possible for all cubic graphs while the seemingly stronger 5-flow Conjecture only applies to bridgeless graphs. We strongly rely on the notion of "k-weak bisections", defined and studied in: L. Esperet, G. Mazzuoccolo and M. Tarsi "Flows and Bisections in Cubic Graphs" J. Graph Theory, 86(2) (2017), 149-158.