Coloring Toeplitz graphs

Let n,a1,a2,…,ak be distinct positive integers. A finite Toeplitz graph Tn(a1,a2,…,ak)=(V,E) is a graph where V={v0,v1,…,vn−1} and E={(vi,vj),for|i−j|∈{a1,a2,…,ak}}. If the number of vertices is infinite, we get an infinite Toeplitz graph. In this paper we first give a complete characterization for connected bipartite finite/infinite Toeplitz graphs. We then focus on finite/infinite Toeplitz graphs with k⩽3, and provide a characterization of their chromatic number.