Graph Puzzles II.1: Counterexamples to Jain's Second Unit Vector Flows Conjecture

TL;DR

This paper constructs counterexamples to Jain's second conjecture on unit vector flows on spheres, challenging assumptions that could imply Tutte's 5-flow conjecture, thus advancing understanding of graph flow properties.

Contribution

The paper provides the first known counterexamples to Jain's second conjecture, demonstrating that the proposed vector assignments cannot always be achieved.

Findings

Counterexamples to Jain's second conjecture are constructed.

Sets of points requiring extended value sets including \

Implications challenge the link to Tutte's 5-flow conjecture.

Abstract

A $3$-dimensional nowhere-zero flow on a graph $G$ is a flow where each edge is assigned a $3$-dimensional vector with unit norm (which corresponds to the points of a $2$-dimensional unit sphere $S^2$). K. Jain posed two conjectures related to this idea. First one suggests that such a flow exists for all bridgeless graphs. The second conjecture states that we can assign values $\{-4,-3,-2,-1,1,2,3,4\}$ to the points of $S^2$, such that antipodal points get opposite values, and values of any three equidistant points on great circles sum to zero. If both conjectures would be true, together they would imply Tutte's 5-flow conjecture. We show 2 counterexamples to the second conjecture, by constructing sets of points each of which additionally requires values $\{-5, 5\}$. Github: https://github.com/gexahedron/unit-vector-flows