Geodesic
A note on forcing 3-repetitions in degree sequences
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 457-462

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In Caro, Shapira and Yuster [Forcing k-repetitions in degree sequences, Electron. J. Combin. 21 (2014) #P1.24] it is proven that for any graph G with at least 5 vertices, one can delete at most 6 vertices such that the subgraph obtained has at least three vertices with the same degree. Furthermore they show that for certain graphs one needs to remove at least 3 vertices in order that the resulting graph has at least 3 vertices of the same degree. In this note we prove that for any graph G with at least 5 vertices, one can delete at most 5 vertices such that the subgraph obtained has at least three vertices with the same degree. We also show that for any triangle-free graph G with at least 6 vertices, one can delete at most one vertex such that the subgraph obtained has at least three vertices with the same degree and this result is tight for triangle-free graphs.
Keywords: repeated degrees 
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Kogan, Shimon. A note on forcing 3-repetitions in degree sequences. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 457-462. http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a9/