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201401 Math 322 Assignment 5
Due: Friday, March 21, 2014, before the lecture starts.
Talking to each other about assigned work is a healthy practice that is encouraged. It
is expected that, in the end, each person’s solution will be written in her own words and
reflect her own understanding.
There are 5 problems, worth a total of 30 marks, plus a bonus problem worth 4 bonus
marks, on two pages.
1. For each of the following, either fill in the missing values to obtain the full collection of parameters (v, b, r, k, λ) of a design or demonstrate that no design with these
parameters can exist.
(a) [2] b = 35, k = 3, λ = 1.
(b) [2] v = 14, b = 7, r = 4.
(c) [2] r = 6, k = 4, λ = 2.
(d) [2] v = 21, b = 28, k = 3.
2. [5] Let v, b, r, k, and λ be integers with k < v, Let A be a v × b 0-1 matrix. Prove that
A is the incidence matrix of a (v, k, λ)-design if and only if AAt = (r − λ)I + λJv×v
and Jv×v A = kJv×b , where Js×t is an s × t matrix with every entry equal to 1.
3. [3] Suppose (V, B) is a (v, k, λ)-design such that no two blocks are the same and not
every k-subset of V is a block. Let B be the collection of k-subsets of V that do not
belong to B. Show that (V, B) is a design and find its parameters.
4. A (v, k, λ)-design is called resolvable if its blocks can be partitioned into r groups (or
parallel classes), each of which is a partition of the point set V.
(a) [2] Show that the condition k | v is necessary for the existence of a resolvable
(v, k, λ)-design and, if this condition and (k − 1) | λ(v − 1) are satisfied, then so
is k(k − 1) | λv(v − 1) (so that the latter conditon is not needed).
(b) [5] Show that if a (v, k, λ)-design is resolvable, then b ≥ v + r − 1.
(Hint: in the incidence matrix A, the collections of columns corresponding to the
parallel classes all have the same sum; now apply some results from linear algebra
to reduce the upper bound on the rank of A .)
5. (a) [2] Make a formal definition of isomorphism of BIBDs D1 = (V1 , B1 ) and D2 =
(V2 , B2 ).
(b) [2] Prove that, up to isomorphism, there is only one (v, 2, 1)-design.
(c) [3] Let D be the (15, 7, 3)-design with V = {0, 1, . . . , 9, A, B, C, D, E} whose blocks
(with brackets and commas omitted) are:
0123456, 012789A, 012BCDE, 03478BC, 0349ADE, 05678DE, 0569ABC, 13579BD,
1367ACE, 1458ABE, 14689CD, 2358ACD, 23689BE, 24579CE, 2467ABD.
Construct the dual design Dd and show that D and Dd are not isomorphic. (Hint:
look at the number of triples of blocks that intersect in three elements.)
6. (Bonus Problem) Suppose a league schedule for 2n teams is to be constructed, with
each pair of teams playing exactly once. (In such a schedule, n games are played
simultaneously on each day the league operates.) It is desirable that each team should
alternate home and away games as much as possible. Define a break in this pattern to
be a repetition of being home, or of being away, in two consecutive games.
(a) [2] Show that the league schedule must contain at least 2n − 2 breaks.
(b) [2] Show how to construct a schedule with exactly 2n − 2 breaks.
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