Coloring-Flow Duality Of Embedded Graphs

Let G be a directed graph embedded in a surface. A map phi : E( G). R is a tension if for every circuit C. G, the sum of f on the forward edges of C is equal to the sum of f 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 phi is a local tension. If 1 <= | phi( e)| <= alpha - 1 holds for every e is an element of 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(c)( G) = inf{alpha is an element of R | G has an alpha-tension} and chi(loc)(G) = inf{alpha is an element of R | G has a local alpha-tension}, respectively. The invariant chi(c) is a refinement of the usual chromatic number, whereas chi(loc) is closely related to Tutte's flow index and Bouchet's biflow index of the surface dual G*.From the definitions we have chi(loc)(G) <= chi(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 X and every epsilon > 0, there exists an integer M so that chi(c)(G) <= chi(loc)(G) + epsilon 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 chi(loc), and thus in chi(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 <= 2r, then chi(c)(G) is an element of [2,2+ epsilon] boolean OR [2r/ r-1, 4]. Similarly, if G is a triangulation with large edge-width, then chi(c)(G) is an element of [3,3+ epsilon]boolean OR[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.