2000 character limit reached
Disconnected Cliques in Derangement Graphs
Published 19 Jul 2024 in math.CO, cs.DM, and math.RT | (2407.14155v1)
Abstract: We obtain a correspondence between pairs of $N\times N$ orthogonal Latin squares and pairs of disconnected maximal cliques in the derangement graph with $N$ symbols. Motivated by methods in spectral clustering, we also obtain modular conditions on fixed point counts of certain permutation sums for the existence of collections of mutually disconnected maximal cliques. We use these modular obstructions to analyze the structure of maximal cliques in $X_N$ for small values of $N$. We culminate in a short, elementary proof of the nonexistence of a solution to Euler's $36$ Officer Problem.
- László Babai. Spectra of Cayley graphs. J. of Combinatorial Theory, Series B, 27(2):180–189, 1979.
- Generating a random permutation with random transpositions. J. Probability Theory and Related Fields, 57:159–179, 1981.
- Steven T. Dougherty. A coding-theoretic solution to the 36363636 officer problem. Des. Codes Cryptogr., 4(2):123–128, 1994.
- The non-existence of maximal sets of four mutually orthogonal Latin squares of order 8. Des. Codes Cryptogr., 33(1):63–69, 2004.
- Enumeration of MOLS of small order. Math. Comp., 85(298):799–824, 2016.
- A. El-Mesady and Shaaban M. Shaaban. Generalization of MacNeish’s Kronecker product theorem of mutually orthogonal latin squares. AKCE International Journal of Graphs and Combinatorics, 18(2):117–122, 2021.
- L. Euler. Recherches sur une nouvelle espece de quarres magiques. Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen, 9:85–239, 1782.
- Martin Gardner. Mathematical games, 1959.
- Gauss. Letter from Schumacher to Gauss, regarding Thomas Clausen. Werke Bd. 12, p. 16, August 10, 1842.
- Pairs of MOLS of order ten satisfying non-trivial relations. Des. Codes Cryptogr., 91(4):1293–1313, 2023.
- Mutually orthogonal latin squares from the inner products of vectors in mutually unbiased bases. Journal of Physics A: Mathematical and Theoretical, 43(13):135302, mar 2010.
- P. Hall. On Representatives of Subsets. J. London Math. Soc., 10(1):26–30, 1935.
- Zhu Z. Chen Y. et. al. Hua, Z. Color image encryption using orthogonal latin squares and a new 2d chaotic system. Nonlinear Dyn, 104:4505–4522, 2021.
- Small Latin squares, quasigroups, and loops. J. Combin. Des., 15(2):98–119, 2007.
- Douglas C. Montgomery. Design and analysis of experiments. Wiley, 10th edition, 2017.
- Integer and constraint programming revisited for mutually orthogonal latin squares, 2021.
- D.R Stinson. A short proof of the nonexistence of a pair of orthogonal latin squares of order six. Journal of Combinatorial Theory, Series A, 36(3):373–376, 1984.
- G. Tarry. Le probléme des 36 Officiers. Compte Rendu de l’Association Française pour l’Avancement des Sciences. Secrétariat de l’Association, 2:170–203, 1901.
- Serge Vaudenay. On the need for multipermutations: Cryptanalysis of md4 and safer. In Bart Preneel, editor, Fast Software Encryption, pages 286–297, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg.
- An application of experimental design using mutually orthogonal latin squares in conformational studies of peptides. Biochemical and biophysical research communications, 316(3):731–737, 2004.
- A short disproof of Euler’s conjecture based on quasi-difference matrices and difference matrices. Discrete Math., 341(4):1114–1119, 2018.
- Harold N. Ward. Thirty-six officers and their code, 2019.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.