Winding number and circular coloring

Abstract

In 1996, Youngs proved a surprising theorem that quadrangulations of the projective plane could never have chromatic number exactly 3. This sparked a lot of interest, and the result has been further developed in many directions over the past decades. For example, the result is strengthened by considering the circular chromatic number, which is a real-valued lower bound on the chromatic number. The circular chromatic number of a quadrangulation cannot be in the interval (2,4). This parameter allows a generalization to larger even faces, for which a similar gap exists. In this work, we place these results into a framework based on the notion of winding number using extensions of colorings to continuous mappings. This yields unified and simplified proofs of gaps in the circular chromatic number for graphs with a distinguished set of directed even cycles. This generalizes the setting of graphs embedded on surfaces where every face is even. We further establish an analogous gap phenomenon when all faces are of a given odd length, previously known only in the case of triangulations. For example, we conclude that if G is a graph embedded on the projective plane such that all faces are 5-cycles, then either its circular chromatic number is 5/2 or at least 3, the former being the case only if G is Eulerian and every noncontractible facial walk is of odd length...