Algebraic Graph Theory Seminar - Akihiro Yamamura (8.11.2024)
Friday 8.11.2024 at 13:15, Lecture room VIII (online too)
Akihiro Yamamura (Akita University Japan):
Latin Regular Hexahedra (Joint work with Haruki Fukaura)
Abstract:
Another generalization of a Latin square [1], called a Latin regular hexahedron, is introduced in [3]. A concrete construction using related combinatorial structures called a Latin three-axis design is given in the paper. It is shown that the construction does not produce all of the Latin regular hexahedra. We examine the construction for Latin regular hexahedra of small order. We call a Latin regular hexahedron defined using Latin three-axis designs separable and inseparable otherwise. We show that all Latin regular hexahedra of order 2 are separable using graph theoretical method, whereas there exists an inseparable Latin regular hexahedra of order 4. We are also interested in how many separable and inseparable Latin regular hexahedra of small order. Using graphs of Latin regular hexahedra and applying Cauchy-Frobenius lemma, we count the number of Latin regular hexahedra of order 2. Then we verify our counting is correct by computer experiment. The investigation concludes that every Latin regular hexahedron of order 2 is separable. The existence of a Latin three-axis design and a Latin four-axis design are equivalent to 1-factorizations [2] of the complete tripartite graph K2n,2n,2n and the complete quadripartite graph Kn,n,n,n, respectively. In addition, we discuss perfect 1-factorization of some of these graphs. In particular, we show that every 1-factorization of K2,2,2 is perfect.
References
[1] J.D´enes and A.D.Keedwell, Latin Squares and their Applications New Developments in the Theory and Applications, 2nd Edition, Elsevier, (2015).
[2] W.Wallis, One-Factorizations, Kluwer Academic Publishers, (1997).
[3] A.Yamamura, Latin Hexahedra and Related Combinatorial Structures, Lecture Notes in Computer Science, Vol. 13947, (2023), 351–362
Those of you who are not able to attend in person or who are still uncertain about the safety of attending in person are welcome to attend via MS Teams. In either case, we hope to see as many of you as possible (either in person or virtually) at our Friday gatherings.