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Steiner trigraphical designs and a block -size bound
Abstract
A primary problem in combinatorial design theory is to determine when designs exist with prescribed properties. Here we focus on proper Steiner t-wise balanced designs. A proper Steiner t-wise balanced design (tBD) of type t-(v, [special characters omitted], 1) is a pair ([special characters omitted], [special characters omitted]) where [special characters omitted] is a v-element set of points, [special characters omitted] is a subset of integers strictly between t and v, and [special characters omitted] is a collection of subsets of [special characters omitted], called blocks, with the property that the size of every block is in [special characters omitted] and every t-element subset of [special characters omitted] is contained in exactly one block. For t ≥ 2, it is conjectured that if t is even then the maximum block size in a proper Steiner tBD is (v-1)/2 and if t is odd the maximum block size is v/2. This conjecture has previously been proven for t = 2, 3, 4 and 5. We prove that a proper Steiner tBD for t = 6 has a maximum block-size of v/2. We then attempt to determine all proper Steiner trigraphical tBD's. Here the v points of our design are the n3 triangles contained in the complete tri-partite graph Kn,n,n and the automorphism group of Kn,n,n, and of our design, is the wreath product Sn wr S3. There are exactly seven designs in the range 1 ≤ t ≤ 3 and no designs in the range 4 ≤ t ≤ 16. We conjecture that there are no proper Steiner trigraphical tBD's for t ≥ 4. We establish constraints on the size of n given t.
Subject Area
Mathematics
Recommended Citation
Ira, Michael S, "Steiner trigraphical designs and a block -size bound" (2000). ETD collection for University of Nebraska-Lincoln. AAI9967378.
https://digitalcommons.unl.edu/dissertations/AAI9967378