Group Irregularity Strength of Connected Graphs

TL;DR

This paper determines the exact group irregularity strength for connected graphs, showing it equals the number of vertices except for specific cases involving stars and certain orders.

Contribution

It establishes precise values of the group irregularity strength for all connected graphs of order at least 3, refining previous bounds and identifying exceptional cases.

Findings

For connected graphs of order at least 3, s_g(G)=n unless n=4k+2.

s_g(G) is at most n+1 in the exceptional cases.

Identifies infinite families of star graphs with different irregularity strength.

Abstract

We investigate the group irregularity strength ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph $G$ of order at least 3, $s_g(G)=n$ if $n\neq 4k+2$ and $s_g(G)\leq n+1$ otherwise, except the case of some infinite family of stars.