# Transversal system

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transversal design, transversal scheme, $T$-system

A system $T _ {0} ( m, t)$ of sets defined for a given collection of $m$ pairwise-disjoint finite sets $S _ {1}, \dots, S _ {m}$, each of which has cardinality $t$. Namely: a transversal system $T _ {0} ( m, t)$ is a system of $t ^ {2}$ sets $Y _ {1}, \dots, Y _ {t ^ {2} }$ (blocks or transversals), containing $m$ elements each and such that:

1) $| Y _ {j} \cap S _ {i} | = 1$; $i = 1, \dots, m$; $j = 1, \dots, t ^ {2}$;

2) $| Y _ {j} \cap Y _ {k} | \leq 1$ for $j \neq k$.

In a transversal design, any two elements $a \in S _ {i}$ and $b \in S _ {j}$, $i \neq j$, occur together in exactly one block. The existence of a transversal design $T _ {0} ( m, t)$ is equivalent to the existence of an orthogonal array $\mathop{\rm OA} ( t, m)$.

Transversal designs are used in recursive constructions of block designs (cf. Block design).

A set of $t$ transversals in $T _ {0} ( m, t)$ is called parallel if no two of them intersect. If a transversal design $T _ {0} ( m, t)$ contains $e$ (or more) parallel classes, then it is denoted by $T _ {e} ( m, t)$.

Some of the basic properties of transversal systems are:

a) if $T _ {d} ( m, s)$ and $T _ {e} ( m, t)$ exist, then so does $T _ {de} ( m, st)$;

b) $T _ {t} ( m - 1, t)$ exists if and only if $T _ {0} ( m, t)$ exists.

#### References

 [1] M. Hall, "Combinatorial theory" , Wiley (1986) [2] H. Hanani, "The existence and construction of balanced incomplete block designs" Ann. Math. Stat. , 32 (1961) pp. 361–386

The finite sets $S _ {1}, \dots, S _ {m}$ making up the design are called point classes or point groups.

The existence of a transversal design $T _ {0} ( m, t)$ is equivalent to the existence of $m - 2$ mutually orthogonal Latin squares of order $t$. If a $T _ {0} ( m, t)$ exists, and $t > 1$, then $m \leq t + 1$.

See also [a1][a3] for the recursive construction of and existence results for transversal designs (and of their generalization "transversal design with holes" , [a2]).

One of the most important results on recursive construction of transversal designs (due to R.M. Wilson [a4], see also [a1]) is as follows:

Let $D$ be a $T _ {0} ( k+ m, s)$ with point classes $P _ {1}, \dots, P _ {k}$ and $Q _ {1}, \dots, Q _ {m}$, let $T$ be a $t$-subset of $Q _ {1} \cup {} \dots \cup Q _ {m}$ and put $t _ {i} = | T \cap Q _ {i} |$ for $i = 1, \dots, m$, and $u _ {B} = | B \cap T |$ for every block $B$ of $D$. Assume the existence of $T _ {0} ( k , t _ {i} )$ for $i = 1, \dots, m$ and of $T _ {u _ {B} } ( k , n + u _ {B} )$ for each block $B$ of $D$. Then $T _ {0} ( k, ns+ t )$ also exists.

Transversals designs (and their dual structures, nets) are also of considerable geometric and algebraic interest. For instance, a $T _ {0} ( s+ 1, s)$ is equivalent to an affine or projective plane of order $s$, and a $T _ {0} ( 3, s)$ is basically the same as a quasi-group of order $s$. Thus, the geometric and algebraic properties of transversal designs (including the study of their automorphism groups) have found considerable interest, cf. [a4].

#### References

 [a1] T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986) [a2] A.E. Brouwer, "Recursive constructions of mutually orthogonal Latin squares" Ann. Discr. Math. , 46 (1991) pp. 149–168 [a3] D. Jungnickel, "Latin squares, their geometries and their groups: a survey" D. Ray-Chaudhuri (ed.) , Coding Theory and Design Theory , IMA Vol. Math. Appl. , 21 , Springer (1990) pp. 166–225 [a4] R.M. Wilson, "Concerning the number of mutually orthogonal Latin squares" Discr. Math. , 9 (1974) pp. 181–198 [a5] H. Hanani, "Balanced incomplete block designs and related designs" Discrete Math. , 11 (1975) pp. 255–369
How to Cite This Entry:
Transversal system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transversal_system&oldid=52085
This article was adapted from an original article by V.E. Tarakanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article