  Given a set D of positive integers, the associated distance graph on the
integers is the graph with the integers as vertices and an edge between
distinct vertices if their difference lies in D. We investigate the chromatic
numbers of distance graphs. We show that, if $D = {d_1,d_2,d_3,...}$, with $d_n
| d_{n+1}$ for all n, then the distance graph has a proper 4-coloring. We
further find the exact chromatic numbers of all such distance graphs. Next, we
characterize those distance graphs that have periodic proper colorings and show
a relationship between the chromatic number and the existence of periodic
proper colorings.

Combinatorics