Coloring-flow duality of embedded graphs

Abstract

Let ▫$G$▫ be a directed graph embedded in a surface. A map ▫$\phi: E(G) \rightarrow \mathbb{R}$▫ is a tension if for every circuit ▫${\mathcal C} \subseteq G$▫, the sum of ▫$\phi$▫ on the forward edges of ▫${\mathcal C}$▫ is equal to the sum of ▫$\phi$▫ on the backward edges of ▫${\mathcal C}$▫. If this condition is satisfied for every circuit of ▫$G$▫ which is contractible curve in the surface, then ▫$\phi$▫ is a local tension. If ▫$1 \le |\phi(e)| \le \alpha - 1$▫ holds for every ▫$e \in E(G)$▫, we say that ▫$\phi$▫ is a (local) ▫$\alpha$▫-tension. We define the circular chromatic number and the local circular chromatic number of ▫$G$▫ by ▫$\chi_{\mathcal{C}}(G) = \inf \{ \alpha \in \mathbb{R} \; | \;G \; \textrm{has an} \;\alpha-\textrm{tension}\}$▫ and ▫$\chi_{\textrm{loc}}(G) = \inf \{ \alpha \in \mathbb{R} \; | \; G \; \textrm{has a local} \; \alpha-\textrm{tension}\}$▫, respectively. The invariant ▫$\chi_{\mathcal{C}}$▫ is a refinement of the usual chromatic number, whereas ▫$\chi_{\textrm{loc}}$▫ is closed related to Tutte's flow index and Bouchet's biflow index of the surface dual ▫$G^\ast$▫. From the definitions we have ▫$\chi_{\textrm{loc}}(G) \le \chi_{\mathcal{C}}(G)$▫. The main result of this paper is a far reaching generalization of Tutte's coloring flow-duality in planar graphs. It is proved that for every surface ▫$\mathbb{X}$▫ and every ▫$\varepsilon > 0$▫, there exist an integer ▫$M$▫ so that ▫$\chi_{\mathcal{C}}(G) \le \chi_{\textrm{loc}}(G) + \varepsilon$▫ holds for every graph embedded in ▫$\mathbb{X}$▫ with edge-width at least ▫$M$▫, where the edge-width is the length of a shortest noncontractible circuit in ▫$G$▫. In 1996, Youngs discovered that every quadrangulation of the projective plane has chromatic number 2 or 4, but never 3. As an application of the main result we show that such bimodal behavior can be observed in ▫$\chi_{\textrm{loc}}$▫, and thus in ▫$\chi_{\mathcal{C}}$▫ for two generic classes of embedded graphs: those that are triangulations and those whose face boundaries all have even length. In particular, if ▫$G$▫ is embedded in some surface with large edge width and all its faces have even length ▫$\le 2r$▫, then ▫$\chi_{\mathcal{C}}(G) \in [2,2 + \varepsilon] \cup [\frac{2r}{r-1},4]$▫. Similarly, if ▫$G$▫ is a triangulation with large edge-width, then ▫$\chi_{\mathcal{C}}(G) \in [3,3 + \varepsilon] \cup [4,5]$▫. It is also shown that there exist Eulerian triangulations of arbitrary large edge-width on nonoriantable surfaces whose circular chromatic number is equal to 5.