Coloring-flow Duality of Embedded Graphs

Let G be a directed graph embedded in a surface. A map φ : E(G) → R is a tension if for every circuit C ⊆ G, the sum of φ on the forward edges of C is equal to the sum of φ on the backward edges of C. If this condition is satisfied for every circuit of G which is a contractible curve in the surface, then φ is a local tension. If 1 ≤ |φ(e)| ≤ α − 1 holds for every e ∈ E(G), we say that φ is a (local) α-tension. We define the circular chromatic number and the local circular chromatic number of G by χc(G) = inf{α ∈ R | G has an α-tension} and χloc(G) = inf{α ∈ R | G has a local α-tension}, respectively. The invariant χc is a refinement of the usual chromatic number, whereas χloc is closely related to Tutte’s flow index and Bouchet’s biflow index of the surface dual G. From the definitions we have χloc(G) ≤ χc(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 X and every ε > 0, there exists an integer M so that χc(G) ≤ χloc(G) + ε holds for every graph embedded in 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 χloc, and thus in χc 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 ≤ 2r, then χc(G) ∈ [2, 2+ε]∪ [ 2r r−1 , 4]. Similarly, if G is a triangulation with large edge-width, then χc(G) ∈ [3, 3+ε]∪[4, 5]. It is also shown that there exist Eulerian triangulations of arbitrarily large edge-width on nonorientable surfaces whose circular chromatic number is equal to 5.