Strong 3‐Flow Conjecture for projective planar graphs

Abstract

AbstractIn 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. to allow vertices to have a prescribed, possibly nonzero difference (modulo 3) between the inflow and outflow. They conjectured that all 5‐edge‐connected graphs with a prescription function have a nowhere‐zero 3‐flow meeting that prescription. Kochol showed that replacing 4‐edge‐connected with 5‐edge‐connected would suffice to prove the 3‐Flow Conjecture and Lovász et al. 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 Grötzsch's Colouring Theorem. Steinberg and Younger provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al. 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.