Note on group irregularity strength of disconnected graphs

TL;DR

This paper explores the group irregularity strength of disconnected graphs, providing bounds and exact values, especially for graphs without star components, advancing understanding of labelings in graph theory.

Contribution

It establishes upper bounds for the group irregularity strength of all graphs and determines exact values for disconnected graphs lacking star components.

Findings

Provided upper bounds for all graphs' group irregularity strength.

Determined exact values for disconnected graphs without star components.

Extended the understanding of group labelings in disconnected graphs.

Abstract

We investigate the \textit{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. So far it was not known if $s_g(G)$ is bounded for disconnected graphs. In the paper we we present some upper bound for all graphs. Moreover we give the exact values and bounds on $s_g(G)$ for disconnected graphs without a star as a component.