Bipartite finite Toeplitz graphs

Let $n,a_1, ..., a_k$ be distinct positive integers: a finite Toeplitz graph is a graph with $n$ vertices, two of which are connected by an edge iff the absolute value of the difference of their indices is $a_1$, $a_2$, ..., or $a_k$. We characterize the whole family of bipartite finite Toeplitz graphs with $k=3$: the proved result completes the characterization of their chromatic number, and is based on a simple characterization of bipartite finite Toeplitz graph with $k=2$. In addition, we characterize some classes of bipartite Toeplitz graphs with $k=4$.