Group Irregular Labelings of Disconnected Graphs

TL;DR

This paper studies the group irregularity strength and modular edge gracefulness of disconnected graphs, providing exact values and bounds for these graph labeling parameters.

Contribution

It introduces new bounds and exact values for the group irregularity strength and modular edge gracefulness of specific families of disconnected graphs.

Findings

Exact values and bounds for $s_g(G)$ for certain disconnected graphs.

Results on the modular edge gracefulness $k(G)$ for disconnected graphs.

New theoretical insights into graph labelings with Abelian groups.

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. We give the exact values and bounds on $s_g(G)$ for chosen families of disconnected graphs. In addition we present some results for the \textit{modular edge gracefulness} $k(G)$, i.e. the smallest value of $s$ such that there exists a function $f:E(G)\rightarrow \zet_s$ such that the sums of edge labels at every vertex are distinct.