On Link-irregular labelings of Graphs

TL;DR

This paper introduces link-irregular labelings for graphs, establishing conditions for their existence, calculating the labeling number for specific graph families, and demonstrating that any positive integer can be realized as a labeling number.

Contribution

It defines link-irregular labelings, provides necessary and sufficient conditions for their existence, and computes the labeling number for various graph families.

Findings

Complete graphs have a link-irregular labeling number of 2 for n ≥ 6.

Wheel graphs have a labeling number approximately proportional to the square root of their size.

Any positive integer can be realized as a graph's link-irregular labeling number.

Abstract

We introduce the concept of link-irregular labelings for graphs, extending the notion of link-irregular graphs through edge labeling with positive integers. A labeling is link-irregular if every vertex has a uniquely labeled subgraph induced by its neighbors. We establish necessary and sufficient conditions for the existence of such labelings and define the link-irregular labeling number $\eta(G)$ as the minimum number of distinct labels required. Our main results include necessary and sufficient conditions for the existence of link-irregular labelings. We show that certain families of graphs, such as bipartite graphs, trees, cycles, hypercubes, and complete multipartite graphs, do not admit link-irregular labelings, while complete graphs and wheel graphs do. Specifically, we prove that $\eta(K_n) = 2$ for $n \geq 6$ and $\eta(K_n) = 3$ for $n \in \{3,4,5\}$. For wheel graphs $W_n$, we establish that $\eta(W_n) \approx \sqrt{2n}$ asymptotically. Finally, we prove that for every positive integer $n$, there exists a graph with a link-irregular labeling number exactly $n$, and provide several results on graph operations that preserve labeling numbers.