Block-avoiding point sequencings of directed triple systems

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Department of Mathematical Sciences


A directed triple system of order v (or, DTS(v)) is a decomposition of the complete directed graph K⃗ v into transitive triples. A v-good sequencing of a DTS(v) is a permutation of the points of the design, say [x1 · · · xv ], such that, for every triple (x, y, z) in the design, it is not the case that x = xi , y = xj and z = xk with i < j < k. We prove that there exists a DTS(v) having a v-good sequencing for all positive integers v ≡ 0, 1 mod 3. Further, for all positive integers v ≡ 0, 1 mod 3, v ≥ 7, we prove that there is a DTS(v) that does not have a v-good sequencing. We also derive some computational results concerning v-good sequencings of all the nonisomorphic DTS(v) for v ≤ 7.

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Discrete Mathematics