SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 497-519, January 2020.
For integers $a\ge 2b>0$, a circular $a/b$-flow is a flow that takes values from $\{\pm b, \pm(b+1), \dots, \pm(a-b)\}$. The Planar Circular Flow Conjecture states that every $2k$-edge-connected planar graph admits a circular $(2+\frac{2}{k})$-flow. The cases $k=1$ and $k=2$ are equivalent to the Four Color Theorem and Grötzsch's 3-Color Theorem. For $k\ge 3$, 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, 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.

中文翻译：

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平面图中的循环流

SIAM离散数学杂志，第34卷，第1期，第497-519页，2020年1月。
对于整数$ a \ ge 2b> 0 $，循环$ a / b $ -flow是从$ \ {\ pm b，\ pm（b + 1），\ dots，\ pm（ab）取值的流\} $。平面圆流猜想指出，每个$ 2k $边连接的平面图都允许循环$（2+ \ frac {2} {k}）$流。$ k = 1 $和$ k = 2 $的情况等效于四色定理和格罗茨奇的三色定理。对于$ k \ ge 3 $，猜想仍然是开放的。在这里，当$ k = 4 $和$ k = 6 $时我们取得了进展。我们证明（i）每10个与边连接的平面图都允许圆形的$ 5/2 $流，并且（ii）每16个与边连接的平面图都允许圆形的7/3流。Dvořák和Postle先前已经证明了关于圆形着色的陈述（i）的双重形式[Combinatorica，37（2017），pp.863--886]，但是我们的证明具有以下优点：更短并且避免使用计算机进行案例检查。进一步，它对反对称流动具有新的含义。陈述（ii）特别有趣，因为与Jaeger原始的循环流猜想相反的例子是12边连接的非平面图，该图不接受循环7/3流。因此，（ii）的平面性假设至关重要。