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 prescribed, possibly non-zero difference (modulo $3$) between inflow and outflow. They conjectured $5$-edge-connected with valid prescription function $3$-flow meeting (we call this Strong Conjecture). Kochol (2001) showed replacing would suffice prove Conjecture Lovasz al.(2013) Conjectures hold if edge connectivity condition is relaxed $6$-edge-connected. Both problems are still open for graphs. The known planar graphs, as it dual of Grotzsch's Colouring Theorem. Steinberg Younger (1989) provided first direct proof using flows well projective Richter al.(2016) We provide two extensions their result, we developed in order