A generalization of Connor's inequality to t-designs with automorphisms

AbstractIn this paper the incidence algebra for t-designs with automorphisms and the fundamental theorem discovered in [4] are exploited to obtain a generalization of Connor's inequality

Block-avoiding sequencings of points in Steiner triple systems

Given an STS(v), we ask if there is a permutation of the points of the design such that no l consecutive points in this permutation contain a block of the design. Such a permutation is called an l-good sequenc-ing. We prove that 3-good sequencings exist for any STS(v) with v\u3e3 and 4-good sequencings exist for any STS(v) with v\u3e71. Similar re-sults also hold for partial STS(v). Finally, we determine the existence or nonexistence of 4-good sequencings for all the nonisomorphic STS(v) with v =7, 9, 13 and 15

Block-avoiding sequencings of points in Steiner triple systems

Given an STS(v), we ask if there is a permutation of the points of the design such that no L consecutive points in this permutation contain a block of the design. Such a permutation is called an L-good sequencing. We prove that 3-good sequencings exist for any STS(v) with v\u3e3and 4-good sequencings exist for any STS(v) with v\u3e71. Similar results also hold for partial STS(v). Finally, we determine the existence or nonexistence of 4-good sequencings for all the nonisomorphic STS(v) with v=7,9,13 and 15

The 3-GDDs of type g3u2g^3u^2

A 3-GDD of type ${g^3u^2}$ exists if and only if $g$ and $u$ have the same parity, $3$ divides $u$ and $u\leq 3g$.Such a 3-GDD of type ${g^3u^2}$ is equivalent to an edge decomposition of $K_{g,g,g,u,u}$ into triangles

Bussey systems and Steiner\u27s tactical problem

In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems. However, solutions to Steiner\u27s questions coincide with Steiner systems only for strengths two and three. For larger strengths, essentially only one class of solutions to Steiner\u27s tactical problems is known, found by Bussey more than a century ago. In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner\u27s tactical problem (Bussey systems) are discussed. For the latter, computational results are provided for at most 15 points

Uniformly resolvable decompositions of Kv in 1-factors and 4-stars

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. A uniformly resolvable {X, Y }-decomposition of the complete graph Kv is an edge decomposition of Kv into exactly r X-factors and s Y -factors. In this article we determine necessary and sufficient conditions for when the complete graph Kv has a uniformly resolvable decompositions into 1-factors and K1,4-factors