On vertex in-out-antimagic total digraphs

Abstract

A vertex in-out-antimagic total labeling of a directed graph (digraph) D=(V,A) with n vertices and m arcs is a bijection from the set V∪A to the set of integers {1,2,…,m+n} such that all n vertex in-weights are pairwise distinct and simultaneously all n vertex out-weights are pairwise distinct. The vertex in-weight is the sum of the vertex label and the labels of all incoming arcs and the vertex out-weight is the sum of the vertex label and the labels of all outgoing arcs.In this paper we provide a general way how to label dense digraphs and certain sparse digraphs. Further, we add constructions of the labeling for three large infinite classes of digraphs and we conjecture that all digraphs allow such a labeling.