Lindner's conjecture that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if v ≡ 1, 3 (mod 6) and v ≥ 2u + 1 is proved.

We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the longest cycle is at most three times as long as the second longest. We then use this result to obtain results on incomplete even cycle systems with a hole and on decompositions of complete multipartite graphs into cycles of uniform even length.

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.

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 m1+⋯+mt=MJX-TeXAtom-OPEN(0n2MJX-TeXAtom-CLOSE). 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 m1+⋯+mt=MJX-TeXAtom-OPEN(0n2MJX-TeXAtom-CLOSE)−n/2.

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.

Let m 1 , m 2 , . . . ,m t be a list of integers. It is shown that there exists an integer N such that for all n N, the complete graph of order n can be decomposed into edge-disjoint cycles of lengths m 1 , m 2 , . . . ,m t if and only if n is odd, 3 m i n for i = 1, 2, . . . ,t, and m 1 +m 2 + · · · +m t = n 2 . In 1981, Alspach conjectured that this result holds for all n, and that a corresponding result also holds for decompositions of complete graphs of even order into cycles and a perfect matching.

The problem of decomposing complete graphs into cycles of arbitrary specified lengths has attracted much attention, but remains largely unsolved. In this paper, the problem is settled in the case where the specified cycle lengths are each more than about half the order of the complete graph. The proof is based on a result that modifies certain existing cycle decompositions to produce new ones in which the lengths of two of the cycles are altered.

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