Two Strong $3$-Flow Theorems for Planar Graphs

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 (we call this the Strong $3$-Flow Conjecture). 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 the $3$-Flow and Strong $3$-Flow 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 provide two extensions to their result, that we developed in order to prove the Strong $3$-Flow Conjecture for projective planar graphs.