Circular Flows in Planar Graphs

Abstract

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 Grötzsch'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 7/3-flow. The dual version of statement (i) on circular coloring was previously proved by Dvořák and Postle (Combinatorica 2017), 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. 1 Jaeger [9] showed that if p, q, r, s ∈ Z + and p/q = r/s, then each graph G has a circular p/q-flow if and only if it has a circular r/s-flow. (See [7] for more details.) We use this result implicitly in the present paper.

Density of 3‐critical signed graphs

“…The case k = 1 is simply Grötzsch's theorem [3]; for k = 2, it has been verified by Dvořák and Postle [2] for the girth condition 10; for k = 3, the best-known girth bound of 16 has very recently been achieved by Postle and Smith-Roberge [20]. The same results for k = 2, 3 were independently obtained by Cranston and Li [1] using the notion of flows. The results of [2] and [20] are each proved by establishing lower bounds on the density of C k 2 +1 -critical graphs, as follows.…”

Circular flows in mono‐directed signed graphs

“…-coloring follows from [25]. Alternative proofs of these two results (for k = 2, 3) are given in [4]. For general values of k, the best results follow from the general results on flows mentioned above.…”

“…Similarly, that every planar graph of odd‐girth at least 17 admits a circular $$\frac{7}{3}$$‐coloring follows from [25]. Alternative proofs of these two results (for $$k=2,3$$) are given in [4]. For general values of $$k$$, the best results follow from the general results on flows mentioned above.…”

Circular coloring and fractional coloring in planar graphs

“…For partial results of Conjecture 1.1, Dvořák and Postle [13] showed that every planar graph of girth at least 10 is C 5 -colorable. In [11], by duality from flow results, a simpler proof of Dvořák and Postle's result was obtained, and it was extended to the next case that every planar graph of girth at least 16 is C 7colorable. Independently, Postle and Smith-Roberge [23] also proved that every planar graph of girth at least 16 is C 7 -colorable through the density of C 7 -critical graphs.…”