Gaps in the Chromatic Spectrum of Face-Constrained Plane Graphs

Let G be a plane graph whose vertices are to be colored subject to constraints on some of the faces. There are 3 types of constraints: a C indicates that the face must contain two vertices of a Common color, a D that it must contain two vertices of a Different color and a B that Both conditions must hold simultaneously. A coloring of the vertices of G satisfying the facial constraints is a strict k-coloring if it uses exactly k colors. The chromatic spectrum of G is the set of all k for which G has a strict k-coloring. We show that a set of integers S is the spectrum of some plane graph with face-constraints if and only if S is an interval {s, s + 1, . . . , t} with 1 ≤ s ≤ 4, or S = {2, 4, 5, . . . , t}, i.e. there is a gap at 3.