Group irregularity strength of connected graphs

Abstract

We investigate the group irregularity strength ($$s_g(G)$$sg(G)) of graphs, that is, we find the minimum value of $$s$$s such that for any Abelian group $$\mathcal G $$G of order $$s$$s, there exists a function $$f:E(G)\rightarrow \mathcal G $$f:E(G)→G such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph $$G$$G of order at least $$3$$3, $$s_g(G)=n$$sg(G)=n if $$ne 4k+2$$n≠4k+2 and $$s_g(G)\le n+1$$sg(G)≤n+1 otherwise, except the case of an infinite family of stars. We also prove that the presented labelling algorithm is linear with respect to the order of the graph.