Zero-sum square matrices

Paul Balister; Yair Caro; Cecil Rousseau; Raphael Yuster

doi:10.1006/eujc.2002.0595

Zero-sum square matrices

Paul Balister, Yair Caro, Cecil Rousseau, Raphael Yuster

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer in for which every square matrix of order at least m has a square submatrix of order k which is zero-sum mod p. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that lim sup M(2, k)/k ≤ 4, lim inf M(3, k)/k ≤ 20, and that M(p, k) ≥ k√2/2e exp(1/e)^p/2. Some nontrivial explicit values are also computed.

Original language	English
Pages (from-to)	489-497
Number of pages	9
Journal	European Journal of Combinatorics
Volume	23
Issue number	5
DOIs	https://doi.org/10.1006/eujc.2002.0595
State	Published - Jul 2002

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.1006/eujc.2002.0595

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title = "Zero-sum square matrices",

abstract = "Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer in for which every square matrix of order at least m has a square submatrix of order k which is zero-sum mod p. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that lim sup M(2, k)/k ≤ 4, lim inf M(3, k)/k ≤ 20, and that M(p, k) ≥ k√2/2e exp(1/e)p/2. Some nontrivial explicit values are also computed.",

author = "Paul Balister and Yair Caro and Cecil Rousseau and Raphael Yuster",

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T1 - Zero-sum square matrices

AU - Balister, Paul

AU - Caro, Yair

AU - Rousseau, Cecil

AU - Yuster, Raphael

PY - 2002/7

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N2 - Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer in for which every square matrix of order at least m has a square submatrix of order k which is zero-sum mod p. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that lim sup M(2, k)/k ≤ 4, lim inf M(3, k)/k ≤ 20, and that M(p, k) ≥ k√2/2e exp(1/e)p/2. Some nontrivial explicit values are also computed.

AB - Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer in for which every square matrix of order at least m has a square submatrix of order k which is zero-sum mod p. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that lim sup M(2, k)/k ≤ 4, lim inf M(3, k)/k ≤ 20, and that M(p, k) ≥ k√2/2e exp(1/e)p/2. Some nontrivial explicit values are also computed.

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DO - 10.1006/eujc.2002.0595

M3 - Article

AN - SCOPUS:0036384256

SN - 0195-6698

VL - 23

SP - 489

EP - 497

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 5

ER -

Zero-sum square matrices

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