Papers › On the Regularity, Planarity and Edge Bounds of Link-irregular Graphs
On the Regularity, Planarity and Edge Bounds of Link-irregular Graphs
Alexander Bastien, Omid Khormali
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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 ≤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√(logn)). 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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