Irregularity strength of digraphs

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc labeling of the digraph such that the ordered pairs of the sums of the in-labels and out-labels at each vertex are all distinct. We define the strength ~s(D) of D to be the smallest of the maximum labels used across all irregular labelings. Similar definitions for graphs have been studied extensively and different formulations of digraph irregularity were given in [15], [12]. In the latter, but not the former reference, the measure involves adding additional vertices. Here we continue the study of irregular labelings of digraphs. We give a general lower bound on ~s(D) and determine ~s(D) exactly for tournaments,! directed paths and cycles and the orientation of the path where all vertices have either in-degree 0 or out-degree 0. We also determine the irregularity strength of a union of directed cycles and a union of directed paths, the latter which requires a new result pertaining to finding circuits of given lengths containing prescribed vertices in the complete symmetric digraph with loops.