A Five-color Theorem for Graphs on Surfaces

We prove that if a graph embeds on a surface with all edges suitably short, then the vertices of the graph can be five-colored. The motivation is that a graph embedded with short edges is locally a planar graph and hence should not require many more than four colors. Introduction. It is well known [4, 15] that a graph embedded on a surface of genus k > 0, the sphere with k handles, can always be //(/c)-colored, where H(k) = [(7 + \/48fc + 1 )/2]; H(k) is called the Heawood number of the surface. A variety of properties are known which ensure that an embedded graph needs significantly fewer than H(k) colors, for example, large girth [12, 13], few triangles (for graphs on the sphere or torus and, more generally, on surfaces of nonnegative Euler characteristic) [9, 11] and Eulerian properties of the (topological) dual graph [10]. On the other hand, one can look for properties which ensure that an embedded graph is locally a planar graph and hence needs not many more than four colors. In this spirit, Mycielski [14] has asked whether for every surface S there is an e > 0 such that a graph embedded on S with edges of length less than e can be five-colored. We restate Mycielski's question in terms of an explicit metric and then answer it in the affirmative for all surfaces. Work of Albertson and Stromquist [3] has already settled the case for the torus (k = 1), and we use many of their techniques in our proof. Also in [3] examples due to J. P. Ballantine and S. Fisk are given which show that no similar result for four-colorability is possible for any surface of positive genus. A surface of genus k > 1 can be represented as a 4/c-sided polygon with pairs of sides identified [7, 16, 18]. If a graph is embedded on a surface of genus k > 1, we obtain a representation Gk of G in and on the boundary of the 4/c-gon. Without loss of generality we take the polygon to be a regular Ak-gon with sides of unit length; we call this the standard Ak-gon, Pk. Each edge of G is represented in Gk by one or more arcs in Pk (if an edge crosses the boundary of Pk, it is divided into pieces). As explained in [5, p. 16], we may assume that each arc of Gk is a polygonal arc; then by the length of an edge of an embedded graph G we mean the sum of the lengths of its polygonal arcs in the representation Gk. Thus length is always defined in terms of a fixed representation of the graph on a standard polygon. Our main result is the following. Received by the editors March 30, 1983. Presented at the 806th meeting of the AMS, October 29, 1983. 1980 Mathematics Subject Classification. Primary 05C15, 05C10.