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Graphs and Combinatorics

, Volume 35, Issue 1, pp 287–301 | Cite as

On Cross Parsons Numbers

  • Cheng Yeaw KuEmail author
  • Kok Bin Wong
Original Paper
  • 32 Downloads

Abstract

Let \(F_q\) be the field of size q and SL(nq) be the special linear group of order n over the field \(F_q\). Assume that n is an even integer. Let \({\mathcal {A}}_i\subseteq SL(n,q)\) for \(i=1,2,\dots , k\) and \(\vert {\mathcal {A}}_1\vert =\vert \mathcal A_2\vert =\cdots =\vert {\mathcal {A}}_k\vert =l\). The set \(\{\mathcal A_1,{\mathcal {A}}_2,\dots ,{\mathcal {A}}_k\}\) is called a k-cross (nq)-Parsons set of size l, if for any pair of (ij) with \(i\ne j\), \(A-B\in SL(n,q)\) for all \(A\in {\mathcal {A}}_i\) and \(B\in {\mathcal {A}}_j\). Let m(knq) be the largest integer l for which there is a k-cross (nq)-Parsons set of size l. The integer m(knq) will be called the k-cross (nq)-Parsons numbers. In this paper, we will show that \(m(3,2,q)\le q\). Furthermore, \(m(3,2,q)= q\) if and only if \(q=4^r\) for some positive integer r. We will also show that if n is a multiple of \(q-1\), then \(m(q-1,n,q)\ge q^{\frac{1}{2}n(n-1)}\).

Keywords

Parsons numbers Parsons graphs 

Mathematics Subject Classification

05D05 05C35 

Notes

Acknowledgements

We would like to thank the anonymous referees for the comments and suggestions that helped us make several improvements to this paper. This project is supported by University of Malaya Research Grant GPF025B-2018.

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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  2. 2.Institute of Mathematical SciencesUniversity of MalayaKuala LumpurMalaysia

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