| Abstract: | A proper vertex coloring of a graph G is achromatic (respectively harmonious) if every two colors appear together on at least one (resp. at most one) edge. The largest (resp. the smallest) number of colors in an achromatic (resp. a harmonious) coloring of G is called the achromatic (resp. harmonious chromatic) number of G and denoted by  (resp.  ). For a finite set of positive integers  and a positive integer n, a circulant graph, denoted by  , is an undirected graph on the set of vertices  that has an edge  if and only if either  or  is a member of  (where substraction is computed modulo n). For any fixed set  , we show that  is asymptotically equal to  , with the error term  . We also prove that  is asymptotically equal to  , with the error term  . As corollaries, we get results that improve, for a fixed k, the previously best estimations on the lengths of a shortest k‐radius sequence over an n‐ary alphabet (i.e., a sequence in which any two distinct elements of the alphabet occur within distance k of each other) and a longest packing k‐radius sequence over an n‐ary alphabet (which is a dual counterpart of a k‐radius sequence). |
