### Gaps in the Chromatic Spectrum of Face-Constrained Plane Graphs

#### Abstract

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 $C$ommon color, a $D$ that it must contain two vertices of a $D$ifferent color and a $B$ that $B$oth 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 $\{s,s+1,\dots,t\}$ with $1\leq s\leq 4$, or $S=\{2,4,5,\dots,t\}$, i.e. there is a gap at 3.