Graphs and Combinatorics

, Volume 35, Issue 1, pp 287–301

# On Cross Parsons Numbers

• Cheng Yeaw Ku
• Kok Bin Wong
Original Paper

## 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

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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