TY - JOUR

T1 - Forcing k-repetitions in degree sequences

AU - Caro, Yair

AU - Shapira, Asaf

AU - Yuster, Raphael

PY - 2014/2/7

Y1 - 2014/2/7

N2 - 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.

AB - 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.

KW - Degree sequence

KW - Repetition

UR - http://www.scopus.com/inward/record.url?scp=84893626466&partnerID=8YFLogxK

U2 - 10.37236/3503

DO - 10.37236/3503

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AN - SCOPUS:84893626466

SN - 0022-5282

VL - 21

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

ER -