Modular Edge-Gracefulness of Graphs without Stars

Abstract

We investigate the modular edge-gracefulness k(G) of a graph, i.e., the least integer k such that taking a cyclic group Zk of order k, there exists a function f:E(G)→Zk so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on k(G) for a general graph G is 2n, where n is the order of G. In this note we prove that if G is a graph of order n without star as a component then k(G)=n for n¬≡2(mod4) and k(G)=n+1 otherwise. Moreover we show that for such G for every integer t≥k(G) there exists a Zt-irregular labeling.