Does the term Row-Complete have any synonyms?

Question

I'm wondering if there is other terminology that describes row-completeness, outside the context of a latin square, or if row-complete is actually a general term.

Answer 1

For those who are unfamiliar, a Latin square $L=(l_{ij})$ is an n by n matrix in which each symbol occurs exactly once in each row an each column, such as the following:

0 1 5 2 4 3
1 2 0 3 5 4
5 0 4 1 3 2
2 3 1 4 0 5
4 5 3 0 2 1
3 4 2 5 1 0

A Latin square L is called row-complete if there are n(n-1) distinct tuples of the form $(l_{ij},l_{i(j+1)})$ in L. The above Latin square is a row-complete Latin square (sourced from http://designtheory.org/library/encyc/exs/clsex.html) since if we look through it we find each ordered pair (0,1), (0,2), ..., (5,4) exactly once. These Latin squares (and Latin squares in general) are sometimes used in statistical experiments (or so I'm led to believe).

For which orders n does there exist a row-complete Latin square is still not completely resolved (although some partial results are known). In the abstract of J. T. Higham's PhD thesis Construction methods for row-complete Latin squares, row-complete Latin squares are also known as roman squares. I haven't heard this term previously myself.

I don't know of any cases where "row-complete" has been used in other contexts, and searching through Google and MathSciNet didn't yield anything. There are probably equivalent notions used in graph theory but under a different name.

If you really need an answer to the question, my feeling is you're going to need to ask a bunch of experts in the field. E.g. you could present your work at a seminar or conference, and ask the audience if they know of an alternative name or usage of "row-complete".