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Circular Flows in Planar Graphs
- Source :
- SIAM J. Discrete Math. Vol. 34(1), 2020, pp. 497-519
- Publication Year :
- 2018
-
Abstract
- For integers $a\ge 2b>0$, a \emph{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 Gr\"otzsch'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) {\em every 10-edge-connected planar graph admits a circular 5/2-flow} and (ii) {\em 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 Dvo\v{r}\'ak 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.<br />Comment: 18 pages, 6 figures, plus 2.5 page appendix with 2 figures, comments welcome
- Subjects :
- Mathematics - Combinatorics
05C15
Subjects
Details
- Database :
- arXiv
- Journal :
- SIAM J. Discrete Math. Vol. 34(1), 2020, pp. 497-519
- Publication Type :
- Report
- Accession number :
- edsarx.1812.09833
- Document Type :
- Working Paper