Strong $3$-Flow Conjecture for Projective Planar Graphs

TL;DR

This paper proves the Strong 3-Flow Conjecture for projective planar graphs, extending the known results from planar graphs and contributing to the understanding of flow conjectures in graph theory.

Contribution

It provides the first proof of the Strong 3-Flow Conjecture specifically for projective planar graphs, a significant extension of previous planar graph results.

Findings

Proves the Strong 3-Flow Conjecture for projective planar graphs.

Extends flow conjecture results from planar to projective planar graphs.

Supports the broader 3-Flow Conjecture in more complex graph classes.

Abstract

In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo $3$) between the inflow and outflow. They conjectured that all $5$-edge-connected graphs with a valid prescription function have a nowhere zero $3$-flow meeting that prescription. Kochol (2001) showed that replacing $4$-edge-connected with $5$-edge-connected would suffice to prove the $3$-Flow Conjecture and Lov\'asz et al.(2013) showed that both conjectures hold if the edge connectivity condition is relaxed to $6$-edge-connected. Both problems are still open for $5$-edge-connected graphs. The $3$-Flow Conjecture was known to hold for planar graphs, as it is the dual of Gr\"otzsch's Colouring Theorem. Steinberg and Younger (1989) provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al.(2016) provided the first direct proof using flows of the Strong $3$-Flow Conjecture for planar graphs. We prove the Strong $3$-Flow Conjecture for projective planar graphs.