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, .?} where ? ? n is said to be harmonious coloring of D if for any two edges xy and u? of D, the ordered pair (f(x), f(y)) ? (f(u), f(?)). If the pair (i, i) is not assigned, then / is said to be a proper harmonious coloring of D. The minimum ? is called the proper harmonious coloring number of D. We investigate the proper harmonious coloring number of graphs such as unidirectional paths, unicycles, inspoken (outspoken) wheels, n -ary trees of different levels etc.