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It was proved in 2009 that any partial Steiner triple system of order u has an embedding of order v for each admissible [ABSTRACT FROM AUTHOR]

We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1, …, mt if and only if n is odd, 3≤mi≤n for i=1, …, t, and . We also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1, …, mt if and only if n is even, 3≤mi≤n for i=1, …, t, and . [ABSTRACT FROM PUBLISHER]

Sequential processes can encounter faults as a result of improper ordering of subsets of the events. In order to reveal faults caused by the relative ordering of t or fewer of v events, for some fixed t, a test suite must provide tests so that every ordering of every set of t or fewer events is exercised. Such a test suite is equivalent to a sequence covering array, a set of permutations on v events for which every subsequence of t or fewer events arises in at least one of the permutations. Equivalently it is a (different) set of permutations, a completely t-scrambling set of permutations, in which the images of every set of t chosen events include each of the t! possible "patterns." In event sequence testing, minimizing the number of permutations used is the principal objective. By developing a connection with covering arrays, lower bounds on this minimum in terms of the minimum number of rows in covering arrays are obtained. An existing bound on the largest v for which the minimum can equal t! is improved. A conditional expectation algorithm is developed to generate sequence covering arrays whose number of permutations never exceeds a specified logarithmic function of v when t is fixed, and this method is shown to operate in polynomial time. A recursive product construction is established when t = 3 to construct sequence covering arrays on vw events from ones on v and w events. Finally computational results are given for t ∈{3, 4, 5} to demonstrate the utility of the conditional expectation algorithm and the product construction. [ABSTRACT FROM AUTHOR]