Extending precolorings of subgraphs of locally planar graphs

Abstract

The width of an embedded graph is the length of a shortest noncontractible cycle. Suppose G is embedded in a surface S (either orientable or not) with large width. In this case G is said to be locally planar. Suppose P⊂ V( G) is a set of vertices such that the components of G[ P] are each 2-colorable, have bounded diameter and are suitably distant from each other. We show that any 5-coloring of G[ P] in which each component is 2-colored extends to a 5-coloring of all of G. Thus, for an arbitrary surface, the extension theorems for precolorings of subgraphs of locally planar graphs parallel the results for planar graphs. Crucial to the proof of this result is the nice cycle lemma, viz. If C is a minimal, noncontractible, and nonseparating cycle in a so-called orderly triangulation of at least moderate width, then there is a cycle C′ such that C′ lies within the fourth neighborhood of C, C′ is minimal, homotopic to C, and C′ either has even length or contains a vertex of degree 4. Such a nice cycle is useful in producing 5-colorings. We introduce the idea of optimal shortcuts in order to prove the nice cycle lemma and the idea of relative width in order to prove the main theorem. Our results generalize to extension theorems for precolorings with q≥3 colors.