On Link-irregular labelings of Graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916817551753216 |
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| author | Bastien, Alexander Khormali, Omid |
| author_facet | Bastien, Alexander Khormali, Omid |
| contents | 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 $η(G)$ 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 $η(K_n) = 2$ for $n \geq 6$ and $η(K_n) = 3$ for $n \in \{3,4,5\}$. For wheel graphs $W_n$, we establish that $η(W_n) \approx \sqrt{2n}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_24080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Link-irregular labelings of Graphs Bastien, Alexander Khormali, Omid Combinatorics 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 $η(G)$ 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 $η(K_n) = 2$ for $n \geq 6$ and $η(K_n) = 3$ for $n \in \{3,4,5\}$. For wheel graphs $W_n$, we establish that $η(W_n) \approx \sqrt{2n}$ 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. |
| title | On Link-irregular labelings of Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.24080 |