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Problem 4.56. Find a difference set for a Steiner triple system on 25 points.

Problem 4.56. Find a difference set for a Steiner triple system on 25 points.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please answer problem 4.56. the other pic is information about topic. 

Definition 4.31. A Steiner triple system is a 2 – (v, 3, 1) design, a collection of triples drawn from a set S so that every pair of points is in one triple. We denote Steiner triple systems as STS(v). Lemma 4.27. A Steiner triple system can be thought of as a partition of the edge set of a complete graph into 3-cycles. Proof: If S is a set of n points so that S is the vertex set of Ky, then a partition of the edge set of K, into 3-cycles creates a collection of triples from S that are the vertices of those 3-cycles. The edge set of this complete graph is the set of all pairs drawn from S. Since the 3-cycles partition the edge set, each pair is in one triple. This satisfies the definition of a 2 – (v, 3, 1) design, and the lemma follows. O Example 4.19 is an example of a way of inducing a partition on the edge set of K7 into triangles. Problem 4.50 also uses triangles, these time two, rotated across K13 to generate a partition of a complete graph into triangles. Lemma 4.28. The number of points in a Steiner triple system is either v = 6m+1 or v = 6m+3. Proof: Let T be the set of triples. Since we are partitioning the edges of K, into triangles, the number of triangles, and triples, is one-third of the number of edges, making |T| = v(v – 1)/6. Consider the triples containing a point a E S. If we remove a from S and the triples, we partition the complete graph on S – {a} into pairs, making v – 1 even so that v must be odd. This tells us that v is 1, 3, or 5 (mod 6). Suppose that v is 6m + 5. Then (6т + 5)(6т + 4) 36m2 + 54m + 20 6|T| but 20 is not a multiple of 6, making |T| take on a non-integer value, which is impossible. We thus have that v = 1,3 (mod 6). ¤
Transcribed Image Text:Definition 4.31. A Steiner triple system is a 2 – (v, 3, 1) design, a collection of triples drawn from a set S so that every pair of points is in one triple. We denote Steiner triple systems as STS(v). Lemma 4.27. A Steiner triple system can be thought of as a partition of the edge set of a complete graph into 3-cycles. Proof: If S is a set of n points so that S is the vertex set of Ky, then a partition of the edge set of K, into 3-cycles creates a collection of triples from S that are the vertices of those 3-cycles. The edge set of this complete graph is the set of all pairs drawn from S. Since the 3-cycles partition the edge set, each pair is in one triple. This satisfies the definition of a 2 – (v, 3, 1) design, and the lemma follows. O Example 4.19 is an example of a way of inducing a partition on the edge set of K7 into triangles. Problem 4.50 also uses triangles, these time two, rotated across K13 to generate a partition of a complete graph into triangles. Lemma 4.28. The number of points in a Steiner triple system is either v = 6m+1 or v = 6m+3. Proof: Let T be the set of triples. Since we are partitioning the edges of K, into triangles, the number of triangles, and triples, is one-third of the number of edges, making |T| = v(v – 1)/6. Consider the triples containing a point a E S. If we remove a from S and the triples, we partition the complete graph on S – {a} into pairs, making v – 1 even so that v must be odd. This tells us that v is 1, 3, or 5 (mod 6). Suppose that v is 6m + 5. Then (6т + 5)(6т + 4) 36m2 + 54m + 20 6|T| but 20 is not a multiple of 6, making |T| take on a non-integer value, which is impossible. We thus have that v = 1,3 (mod 6). ¤
Problem 4.56. Find a difference set for a Steiner triple system on 25 points.
Transcribed Image Text:Problem 4.56. Find a difference set for a Steiner triple system on 25 points.
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