Geodesic
Infinite Latin squares: neighbor balance and orthogonality
Anthony B. Evans1 ; Gage N. Martin2 ; Kaethe Minden3 ; M. A. Ollis3
1Wright State University
2Boston College
3Marlboro College
The electronic journal of combinatorics, Tome 27 (2020) no. 3
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Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table. Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.
DOI : 10.37236/8020
Classification : 05B15

Anthony B. Evans  1   ; Gage N. Martin  2   ; Kaethe Minden  3   ; M. A. Ollis  3

1 Wright State University
2 Boston College
3 Marlboro College 
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     title = {Infinite {Latin} squares: neighbor balance and orthogonality},
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     year = {2020},
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Anthony B. Evans; Gage N. Martin; Kaethe Minden; M. A. Ollis. Infinite Latin squares: neighbor balance and orthogonality. The electronic journal of combinatorics, Tome 27 (2020) no. 3. doi: 10.37236/8020

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