CIRCULAR FLOWS IN PLANAR GRAPHS

For integers a >= 2b > 0, a circular a/b-flow is a flow that takes values from {+b, +/-(b+ 1), ..., +/-(a - b)}. The Planar Circular Flow Conjecture states that every 2k-edge-connected planar graph admits a circular (2 + 2/k)-flow. The cases k = 1 and k = 2 are equivalent to the Four Color Theorem and Grotzsch's 3-Color Theorem. For k >= 3, the conjecture remains open. Here we make progress when k = 4 and k = 6. We prove that (i) every 10-edge-connected planar graph admits a circular 5/2-flow and (ii) every 16-edge-connected planar graph admits a circular 7/3flow. The dual version of statement (i) on circular coloring was previously proved by Dvorak and Postle [Combinatorica, 37 (2017), pp. 863-886], but our proof has the advantages of being much shorter and avoiding the use of computers for case-checking. Further, it has new implications for antisymmetric flows. Statement (ii) is especially interesting because the counterexamples to Jaeger's original Circular Flow Conjecture are 12-edge-connected nonplanar graphs that admit no circular 7/3-flow. Thus, the planarity hypothesis of (ii) is essential.