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arXiv · 2503.21916

On the Regularity, Planarity and Edge Bounds of Link-irregular Graphs

Alexander Bastien·Omid Khormali
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Abstract

A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones Mathematicae. Graph Theory, 45(1) (2025) p.95] conjectured that there exists no regular link-irregular graph. In this paper, we show that the existence of an $r$-regular link irregular graph is very likely for large enough $r$. In particular, we provide a 7-regular link irregular graph on 12 vertices, which serves as a counterexample to the conjecture. Additionally, we prove that no bipartite link-irregular graphs exist, and there are no regular link-irregular graphs on $n$-vertices for $n \leq 9$. Also, we determine upper and lower bounds for the number of edges of link-irregular graphs. Furthermore, we show the minimum number of edges in a link-irregular graph on the $n$ vertices is $Ω(n\sqrt{\log n})$. Finally, we prove that all but finitely many link-irregular graphs are non-planar, and there is no regular link-irregular planar graphs.

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Alexander Bastien, Omid Khormali. 2025-06-12. On the Regularity, Planarity and Edge Bounds of Link-irregular Graphs. https://arxiv.org/abs/2503.21916

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