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Group irregularity strength of disconnected graphs

Sylwia Cichacz, Barbara Krupińska

TL;DR

The paper addresses the problem of determining the group irregularity strength $s_g(G)$ for graphs, including disconnected graphs. It develops a constructive labeling approach based on Skolem partitions of odd-order Abelian groups and edge perturbations along shortest even/odd walks to enforce unique vertex weights. The main results provide exact and near-tight bounds for $s_g(G)$ when components avoid small-star components (e.g., $K_{1,2u+1}$), and extend to graphs with even-star components by bounds of the form $| Gamma|\ge n+K(q_0,q_4)$, along with conjectures about a universal constant $K$. The work also presents concrete instances, such as $s_g(2K_{1,3})=8$, illustrating the method's sharpness and scope.

Abstract

We investigate the \textit{group irregular strength} $(s_g(G))$ of graphs, i.e the smallest value of $s$ such that for any Abelian group $Γ$ of order $s$ exists a function $g\colon E(G) \rightarrow Γ$ such that sums of edge labels at every vertex is distinct. We give results for bound and exact values of $(s_g(G))$ for graphs without small stars as components.

Group irregularity strength of disconnected graphs

TL;DR

The paper addresses the problem of determining the group irregularity strength for graphs, including disconnected graphs. It develops a constructive labeling approach based on Skolem partitions of odd-order Abelian groups and edge perturbations along shortest even/odd walks to enforce unique vertex weights. The main results provide exact and near-tight bounds for when components avoid small-star components (e.g., ), and extend to graphs with even-star components by bounds of the form , along with conjectures about a universal constant . The work also presents concrete instances, such as , illustrating the method's sharpness and scope.

Abstract

We investigate the \textit{group irregular strength} of graphs, i.e the smallest value of such that for any Abelian group of order exists a function such that sums of edge labels at every vertex is distinct. We give results for bound and exact values of for graphs without small stars as components.
Paper Structure (2 sections, 10 theorems, 23 equations, 1 figure)

This paper contains 2 sections, 10 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of the main result

Key Result

Theorem 1.1

Let $G$ be an arbitrary connected graph of order $n\geq 3$. Then

Figures (1)

  • Figure 1: Irregular labelings of $2K_{1,3}$ in groups $\mathbb{Z}_8$, $\mathbb{Z}_2\times \mathbb{Z}_4$ and $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$, respectively.

Theorems & Definitions (11)

  • Theorem 1.1: GrIrStrCon
  • Lemma 1.2: GrIrLaDisc
  • Corollary 1.3: LinearBoundNo0
  • Theorem 1.4: GrIrLaDisc
  • Theorem 2.1: PartAb
  • Lemma 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Theorem 2.6
  • ...and 1 more