Achromatic and Harmonious Colorings of Circulant Graphs

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 Psi(G) (resp. h(G)). For a finite set of positive integers D and a positive integer n, a circulant graph, denoted by C-n(D), is an undirected graph on the set of vertices {0, 1,..., n - 1} that has an edge i j if and only if either i - j or j - i is a member of D (where substraction is computed modulo n). For any fixed set D, we show that Psi(C-n(D)) is asymptotically equal to root 2|D|n, with the error term O(log n). We also prove that h(C-n(D)) is asymptotically equal to root 2|D|n, with the error term O(n(1/4) root log n). 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). (C) 2017 Wiley Periodicals, Inc.