Graph Puzzles II.1: Counterexamples to Jain's Second Unit Vector Flows Conjecture
Nikolay Ulyanov
math.CO
Mar 24, 2026 · v1
TL;DR
The counterexample constructions, which force labels ±5 on finite point sets of the sphere, are verified in Lean 4, with code released on GitHub.
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-flowsProblem
K. Jain conjectured that the points of S^2 can be labeled with values in {±1,±2,±3,±4} so that antipodal points get opposite labels and any three equidistant points on a great circle sum to zero. Together with the unit vector flows conjecture, this would imply Tutte's 5-flow conjecture.
Approach
The authors first relate the icosidodecahedron and the Petersen graph to a valid nz5-flow labeling. They then construct finite point subsets of S^2, including a 50-point expansion of the icosidodecahedron. They check these sets with a SAT encoding and backtracking search, cross-validated with exact arithmetic. The constructions are also verified in Lean 4.
Results
Two finite point sets are given that require labels ±5, which disproves Jain's second conjecture. A nowhere-zero 6-flow suffices for them, but a 5-flow does not. Code and the Lean 4 verification are on GitHub.
