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

Figures (2)

• Figure 1: First counterexample to Jain's second conjecture, which might also be reinterpreted as a combination of Petersen graph and Möbius ladder; scaled to radius 2
• Figure 2: Icosidodecahedron mapping for Petersen graph, scaled to radius 2