Forcing $k$-repetitions in degree sequences

Yair Caro, Asaf Shapira, Raphael Yuster

Published 2013-12-04Version 1

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is ``close'' to a graph $G'$ with $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree. Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.