Further Results on Harmonious Colorings of Digraphs

Abstract

Let D be a directed graph with n vertices and m edges. A function f: V (D) ? {1, 2, 3, ..., t}, where t ? n is said to be a harmonious coloring of D if for any two edges xy and uv of D, the ordered pair (f(x), f(y)) ? (f(u), f(v)). If no pair (i, i) is assigned, then f is said to be a proper harmonious coloring of D. The minimum t for which D admits a proper harmonious coloring is called the proper harmonious coloring number of D. We investigate the proper harmonious coloring number of graphs such as alternating paths and alternating cycles.