Coloring Toeplitz graphs

Let n , a 1 , a 2 , … , a k be distinct positive integers. A finite Toeplitz graph T n ( a 1 , a 2 , … , a k ) = ( V , E ) is a graph where V = { v 0 , v 1 , … , v n − 1 } and E = { ( v i , v j ) , for | i − j | ∈ { a 1 , a 2 , … , a k } } . 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.