A note on forcing 3-repetitions in degree sequences

Abstract

In Caro, Shapira and Yuster [1] 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.