Coloring Planar Toeplitz Graphs

Abstract

We study structural properties of infinite Toeplitz graphs such as connectivity, planarity and colorability. As to connectivity we show that any infinite Toeplitz graph decomposes into a finite number of connected and isomorphic components. Similar to the bipartite case (cf. [4]), infinite planar Toeplitz graphs can be characterized by a simple condition on their defining 0-1 sequence. We then turn to the problem of coloring such planar graphs. Whereas they can always be 4-colored by a greedy-like algorithm, we are able to fully characterize the subclass of 3-colorable such graphs. As a corollary we obtain a Konig-type characterization of this class by means of (K₄\e)-cycles, a family of 4-critical graphs that generalize odd cycles. We also discuss some consequences for general (finite) graph coloring.