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For positive integers $m$ and $n$, the Zarankiewicz number $Z_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with $m$ vertices and $n$ edges. Guy determined $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3+O(m)$. Here, we extend this by determining $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3$ and, when $m$ is large, for all $n \geq \binom{m}{2}/6+O(m)$., Comment: 18 pages, 4 figures

In situations where every item in a data set must be compared with every other item in the set, it may be desirable to store the data across a number of machines in such a way that any two data items are stored together on at least one machine. One way to evaluate the efficiency of such a distribution is by the largest fraction of the data it requires to be allocated to any one machine. The all-to-all comparison (ATAC) data limit for $m$ machines is a measure of the minimum of this value across all possible such distributions. In this paper we further the study of ATAC data limits. We observe relationships between them and the previously studied combinatorial parameters of fractional matching numbers and covering numbers. We also prove a lower bound on the ATAC data limit that improves on one of Hall, Kelly and Tian, and examine the special cases where equality in this bound is possible. Finally, we investigate the data limits achievable using various classes of combinatorial designs. In particular, we examine the cases of transversal designs and projective Hjelmslev planes., Comment: 16 pages, 1 figure

Let $\mathcal{F}$ be a family of subsets of a ground set $\{1,\ldots,n\}$ with $|\mathcal{F}|=m$, and let $\mathcal{F}^{\updownarrow}$ denote the family of all subsets of $\{1,\ldots,n\}$ that are subsets or supersets of sets in $\mathcal{F}$. Here we determine the minimum value that $|\mathcal{F}^{\updownarrow}|$ can attain as a function of $n$ and $m$. This can be thought of as a `two-sided' Kruskal-Katona style result. It also gives a solution to the isoperimetric problem on the graph whose vertices are the subsets of $\{1,\ldots,n\}$ and in which two vertices are adjacent if one is a subset of the other. This graph is a supergraph of the $n$-dimensional hypercube and we note some similarities between our results and Harper's theorem, which solves the isoperimetric problem for hypercubes. In particular, analogously to Harper's theorem, we show there is a total ordering of the subsets of $\{1,\ldots,n\}$ such that, for each initial segment $\mathcal{F}$ of this ordering, $\mathcal{F}^{\updownarrow}$ has the minimum possible size. Our results also answer a question that arises naturally out of work of Gerbner et al. on cross-Sperner families and allow us to strengthen one of their main results., Comment: 21 pages, 1 figure

For positive integers $s$, $t$, $m$ and $n$, the Zarankiewicz number $Z_{s,t}(m,n)$ is defined to be the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no complete biparitite subgraph containing $s$ vertices in the part of size $m$ and $t$ vertices in the part of size $n$. A simple argument shows that, for each $t \geq 2$, $Z_{2,t}(m,n)=(t-1)\binom{m}{2}+n$ when $n \geq (t-1)\binom{m}{2}$. Here, for large $m$, we determine the exact value of $Z_{2,t}(m,n)$ in almost all of the remaining cases where $n=\Theta(tm^2)$. We establish a new family of upper bounds on $Z_{2,t}(m,n)$ which complement a family already obtained by Roman. We then prove that the floor of the best of these bounds is almost always achieved. We also show that there are cases in which this floor cannot be achieved and others in which determining whether it is achieved is likely a very hard problem. Our results are proved by viewing the problem through the lens of linear hypergraphs and our constructions make use of existing results on edge decompositions of dense graphs., Comment: 26 pages, 1 figure

A partial $(n,k,t)_\lambda$-system is a pair $(X,\mathcal{B})$ where $X$ is an $n$-set of vertices and $\mathcal{B}$ is a collection of $k$-subsets of $X$ called blocks such that each $t$-set of vertices is a subset of at most $\lambda$ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of $\{0,\ldots,n-1\}$. A sequencing is $\ell$-block avoiding or, more briefly, $\ell$-good if no block is contained in a set of $\ell$ vertices with consecutive labels. Here we give a short proof that, for fixed $k$, $t$ and $\lambda$, any partial $(n,k,t)_\lambda$-system has an $\ell$-good sequencing for some $\ell=\Theta(n^{1/t})$ as $n$ becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case $k=t+1$ where results of Kostochka, Mubayi and Verstra\"{e}te show that the value of $\ell$ cannot be increased beyond $\Theta((n \log n)^{1/t})$. A special case of our result shows that every partial Steiner triple system (partial $(n,3,2)_1$-system) has an $\ell$-good sequencing for each positive integer $\ell \leq 0.0908\,n^{1/2}$., Comment: 9 pages, 0 figures

A $k$-star is a complete bipartite graph $K_{1,k}$. For a graph $G$, a $k$-star decomposition of $G$ is a set of $k$-stars in $G$ whose edge sets partition the edge set of $G$. If we weaken this condition to only demand that each edge of $G$ is in at most one $k$-star, then the resulting object is a partial $k$-star decomposition of $G$. An embedding of a partial $k$-star decomposition $\mathcal{A}$ of a graph $G$ is a partial $k$-star decomposition $\mathcal{B}$ of another graph $H$ such that $\mathcal{A} \subseteq \mathcal{B}$ and $G$ is a subgraph of $H$. This paper considers the problem of when a partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ for a given integer $s$. We improve a result of Noble and Richardson, itself an improvement of a result of Hoffman and Roberts, by showing that any partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ for some $s$ such that $s < \frac{9}{4}k$ when $k$ is odd and $s < (6-2\sqrt{2})k$ when $k$ is even. For general $k$, these constants cannot be improved. We also obtain stronger results subject to placing a lower bound on $n$., Comment: 17 pages, 0 figures

Column replacement techniques for creating covering arrays rely on the construction of perfect and distributing hash families with few rows, having as many columns as possible for a specified number of symbols. To construct distributing hash families in which the number of rows is less than the strength, we examine a method due to Blackburn and extend it in three ways. First, the method is generalized from homogeneous hash families (in which every row has the same number of symbols) to heterogeneous ones. Second, the extension treats distributing hash families, in which only separation into a prescribed number of parts is required, rather than perfect hash families, in which columns must be completely separated. Third, the requirements on one of the main ingredients are relaxed to permit the use of a large class of distributing hash families, which we call fractal. Constructions for fractal perfect and distributing hash families are given, and applications to the construction of perfect hash families of large strength are developed., Comment: 21 pages, 0 figures

When $k$ factors each taking one of $v$ levels may affect the correctness or performance of a complex system, a test is selected by setting each factor to one of its levels and determining whether the system functions as expected (passes the test) or not (fails). In our setting, each test failure can be attributed to at least one faulty (factor, level) pair. A nonadaptive test suite is a selection of such tests to be executed in parallel. One goal is to minimize the number of tests in a test suite from which we can determine which (factor, level) pairs are faulty, if any. In this paper, we determine the number of tests needed to locate faults when exactly one (or at most one) pair is faulty. To do this, we address an equivalent problem, to determine how many set partitions of a set of size $N$ exist in which each partition contains $v$ classes and no two classes in the partitions are equal., Comment: 16 pages, 0 figures

An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set such that each partition has $k$ nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this paper we introduce a new construction for Sperner partition systems based on a division of the ground set into many equal-sized parts. We use this to asymptotically determine $\mathrm{SP}(n,k)$ in many cases where $\frac{n}{k}$ is bounded as $n$ becomes large. Further, we show that this construction produces a Sperner partition system of maximum size for numerous small parameter sets $(n,k)$. By extending a separate existing construction, we also establish the asymptotics of $\mathrm{SP}(n,k)$ when $n \equiv k \pm 1 \pmod{2k}$ for almost all odd values of $k$., Comment: 24 pages, 0 figures

In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fra\"{\i}ss\'{e} limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fra\"{\i}ss\'e's construction and its variants are rich sources of examples that are central to model-theoretic classification theory, and recently infinite Steiner systems obtained via Fra\"{\i}ss\'e-type constructions have received attention from the model theory community., Comment: 15 pages, 1 figure

For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur together in exactly one block. If we weaken this condition to demand only that each pair of points occur together in at most one block, then the resulting object is a partial $(n,k,1)$-design. A completion of a partial $(n,k,1)$-design $(V,\mathcal{A})$ is a (complete) $(n,k,1)$-design $(V,\mathcal{B})$ such that $\mathcal{A} \subseteq \mathcal{B}$. Here, for all sufficiently large $n$, we determine exactly the minimum number of blocks in an uncompletable partial $(n,k,1)$-design. This result is reminiscent of Evans' now-proved conjecture on completions of partial latin squares. We also prove some related results concerning edge decompositions of almost complete graphs into copies of $K_k$., Comment: 17 pages, 0 figures

Nov\'{a}k conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic $(v,k,\lambda)$-designs with $1 \leq \lambda \leq k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime and $\lambda=1$ and also when $\lambda \leq (k-1)/2$ and $v$ is sufficiently large compared to $k$. As a corollary, we show that for any $k \geq 3$, with the possible exception of finitely many composite orders $v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-disjoint difference family., Comment: 9 pages, 0 figures

The cyclic matching sequenceability of a simple graph $G$, denoted $\mathrm{cms}(G)$, is the largest integer $s$ for which there exists a cyclic ordering of the edges of $G$ so that every set of $s$ consecutive edges forms a matching. In this paper we consider the minimum cyclic matching sequenceability of $k$-regular graphs. We completely determine this for $2$-regular graphs, and give bounds for $k \geq 3$., Comment: 24 pages, 1 figure

A partial Steiner triple system of order $u$ is a pair $(U,\mathcal{A})$ where $U$ is a set of $u$ elements and $\mathcal{A}$ is a set of triples of elements of $U$ such that any two elements of $U$ occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system $(U,\mathcal{A})$ is a (complete) Steiner triple system $(V,\mathcal{B})$ such that $U \subseteq V$ and $\mathcal{A} \subseteq \mathcal{B}$. For a given partial Steiner triple system of order $u$ it is known that an embedding of order $v \geq 2u+1$ exists whenever $v$ satisfies the obvious necessary conditions. Determining whether "small" embeddings of order $v < 2u+1$ exist is a more difficult task. Here we extend a result of Colbourn on the $\mathsf{NP}$-completeness of these problems. We also exhibit a family of counterexamples to a conjecture concerning when small embeddings exist., Comment: 11 pages, 2 figures

A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set $V$ generates a digraph with vertex set $V$ and arcs $(x,x^\sigma)$ for $x\in V$ and $\sigma\in\mathcal{S}$. We address the problem of characterising those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erd\H{o}s for vertex colourings of graphs in that the property for an infinite digraph is determined by properties of its finite sub-digraphs. The derangement generation property for a digraph is linked with the existence of a finite $1$-factor cover for an associated bipartite (undirected) graph., Comment: 14 pages, 4 figures

We prove that, for sufficiently large $n$, every graph of order $n$ with minimum degree at least $0.852n$ has a fractional edge-decomposition into triangles. We do this by refining a method used by Dross to establish a bound of $0.9n$. By a result of Barber, K\"{u}hn, Lo and Osthus, our result implies that, for each $\epsilon >0$, every graph of sufficiently large order $n$ with minimum degree at least $(0.852+\epsilon)n$ has a triangle decomposition if and only if it has all even degrees and number of edges a multiple of three., Comment: 15 pages, 0 figures

An $(n,k)$-Sperner partition system is a collection of partitions of some $n$-set, each into $k$ nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this paper we introduce a new construction for Sperner partition systems and use it to asymptotically determine $\mathrm{SP}(n,k)$ in many cases as $\frac{n}{k}$ becomes large. We also give a slightly improved upper bound for $\mathrm{SP}(n,k)$ and exhibit an infinite family of parameter sets $(n,k)$ for which this bound is tight., Comment: 20 pages, 2 figures

An induced path factor of a graph $G$ is a set of induced paths in $G$ with the property that every vertex of $G$ is in exactly one of the paths. The induced path number $\rho(G)$ of $G$ is the minimum number of paths in an induced path factor of $G$. We show that if $G$ is a connected cubic graph on $n>6$ vertices, then $\rho(G)\le(n-1)/3$. Fix an integer $k\ge3$. For each $n$, define $\mathcal{M}_n$ to be the maximum value of $\rho(G)$ over all connected $k$-regular graphs $G$ on $n$ vertices. As $n\rightarrow\infty$ with $nk$ even, we show that $c_k=\lim(\mathcal{M}_n/n)$ exists. We prove that $5/18\le c_3\le1/3$ and $3/7\le c_4\le1/2$ and that $c_k=\frac12-O(k^{-1})$ for $k\rightarrow\infty$., Comment: 20 pages, 3 figures

In 1979 Tarsi showed that an edge decomposition of a complete multigraph into stars of size $m$ exists whenever some obvious necessary conditions hold. In 1992 Lonc gave necessary and sufficient conditions for the existence of an edge decomposition of a (simple) complete graph into stars of sizes $m_1,\ldots,m_t$. We show that the general problem of when a complete multigraph admits a decomposition into stars of sizes $m_1,\ldots,m_t$ is $\mathsf{NP}$-complete, but that it becomes tractable if we place a strong enough upper bound on $\max(m_1,\ldots,m_t)$. We determine the upper bound at which this transition occurs. Along the way we also give a characterisation of when an arbitrary multigraph can be decomposed into stars of sizes $m_1,\ldots,m_t$ with specified centres, and a generalisation of Landau's theorem on tournaments., Comment: 30 pages, 0 figures

For a fixed positive integer $k$, a set $S$ of vertices of a graph or multigraph is called a $k$-independent set if the subgraph induced by $S$ has maximum degree less than $k$. The well-known algorithm MAX finds a maximal $k$-independent set in a graph or multigraph by iteratively removing vertices of maximum degree until what remains has maximum degree less than $k$. We give an efficient procedure that determines, for a given degree sequence $D$, the smallest cardinality $b(D)$ of a $k$-independent set that can result from any application of MAX to any loopless multigraph with degree sequence $D$. This analysis of the worst case is sharp for each degree sequence $D$ in that there exists a multigraph $G$ with degree sequence $D$ such that some application of MAX to $G$ will result in a $k$-independent set of cardinality exactly $b(D)$., Comment: 16 pages, 5 figures

Fisher proved in 1940 that any $2$-$(v,k,\lambda)$ design with $v>k$ has at least $v$ blocks. In 1975 Ray-Chaudhuri and Wilson generalised this result by showing that every $t$-$(v,k,\lambda)$ design with $v \geq k+\lfloor t/2 \rfloor$ has at least $\binom{v}{\lfloor t/2 \rfloor}$ blocks. By combining methods used by Bose and Wilson in proofs of these results, we obtain new lower bounds on the size of $t$-$(v,k,\lambda)$ coverings. Our results generalise lower bounds on the size of $2$-$(v,k,\lambda)$ coverings recently obtained by the first author., Comment: 19 pages, 0 figures

It is conjectured that every Steiner triple system of order $v \neq 7$ has chromatic index at most $(v+3)/2$ when $v \equiv 3 \pmod{6}$ and at most $(v+5)/2$ when $v \equiv 1 \pmod{6}$. Herein, we construct a Steiner triple system of order $v$ with chromatic index at least $(v+3)/2$ for each integer $v \equiv 3 \pmod{6}$ such that $v \geq 15$, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in $v$. Finally, we establish for each order $v \equiv 15 \pmod{18}$ that there are at least $v^{v^2(1/6+o(1))}$ non-isomorphic Steiner triple systems with chromatic index at least $(v+3)/2$ and that some of these systems are cyclic., Comment: 10 pages, 0 figures

We prove that the complete graph with a hole $K_{u+w}-K_u$ can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, each cycle has length at most $\min(u,w)$, and the longest cycle is at most three times as long as the second longest. This generalises existing results on decomposing the complete graph with a hole into cycles of uniform length, and complements work on decomposing complete graphs, complete multigraphs, and complete multipartite graphs into cycles of arbitrary specified lengths., Comment: 41 pages, 0 figures. arXiv admin note: text overlap with arXiv:1411.3785

We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

A $(v,k,\lambda)$-covering is a pair $(V, \mathcal{B})$, where $V$ is a $v$-set of points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ (called blocks), such that every unordered pair of points in $V$ is contained in at least $\lambda$ blocks in $\mathcal{B}$. The excess of such a covering is the multigraph on vertex set $V$ in which the edge between vertices $x$ and $y$ has multiplicity $r_{xy}-\lambda$, where $r_{xy}$ is the number of blocks which contain the pair $\{x,y\}$. A covering is symmetric if it has the same number of blocks as points. Bryant et al.(2011) adapted the determinant related arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the nonexistence of certain symmetric coverings with $2$-regular excesses. Here, we adapt the arguments related to rational congruence of matrices and show that they imply the nonexistence of some cyclic symmetric coverings and of various symmetric coverings with specified excesses., Comment: Submitted on May 22, 2015 to the Journal of Linear Algebra and its Applications

In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the performance of matrices obtained from this construction. We provide new theoretical results and detailed simulations. These simulations indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given., Comment: 18 pages, 3 figures

Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2p where p = 5 (mod 8) is prime.

For each odd $m \geq 3$ we completely solve the problem of when an $m$-cycle system of order $u$ can be embedded in an $m$-cycle system of order $v$, barring a finite number of possible exceptions. In cases where $u$ is large compared to $m$, where $m$ is a prime power, or where $m \leq 15$, the problem is completely resolved. In other cases, the only possible exceptions occur when $v-u$ is small compared to $m$. This result is proved as a consequence of a more general result which gives necessary and sufficient conditions for the existence of an $m$-cycle decomposition of a complete graph of order $v$ with a hole of size $u$ in the case where $u \geq m-2$ and $v-u \geq m+1$ both hold., Comment: 28 pages, 0 figures

In 1940 Fisher famously showed that if there exists a non-trivial $(v,k,\lambda)$-design then $\lambda(v-1) \geq k(k-1)$. Subsequently Bose gave an elegant alternative proof of Fisher's result. Here, we show that the idea behind Bose's proof can be generalised to obtain new bounds on the number of blocks in $(v,k,\lambda)$-coverings and -packings with $\lambda(v-1)

We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to $3 \pmod{6}$. The only previously known examples have order $15$ or $21$., Comment: 5 pages, 0 figures. This version fixes a minor error in the journal version of the paper where we had neglected to say that two sum-zero triples must be removed before the additional triples in $\mathcal{B}^\infty \cup \mathcal{B}^*$ are added to create the Steiner triple system

A combinatorial approach to compressive sensing based on a deterministic column replacement technique is proposed. Informally, it takes as input a pattern matrix and ingredient measurement matrices, and results in a larger measurement matrix by replacing elements of the pattern matrix with columns from the ingredient matrices. This hierarchical technique yields great flexibility in sparse signal recovery. Specifically, recovery for the resulting measurement matrix does not depend on any fixed algorithm but rather on the recovery scheme of each ingredient matrix. In this paper, we investigate certain trade-offs for signal recovery, considering the computational investment required. Coping with noise in signal recovery requires additional conditions, both on the pattern matrix and on the ingredient measurement matrices., Comment: 11 pages, 6 figures

It was proved in 2009 that any partial Steiner triple system of order $u$ has an embedding of order $v$ for each admissible integer $v\geq 2u+1$. This result is best-possible in the sense that, for each $u\geq 9$, there exists a partial Steiner triple system of order $u$ that does not have an embedding of order $v$ for any $v<2u+1$. Many partial Steiner triple systems do have embeddings of orders smaller than $2u+1$, but little has been proved about when these embeddings exist. In this paper we construct embeddings of orders less than $2u+1$ for partial Steiner triple systems with few triples. In particular, we show that a partial Steiner triple system of order $u \geq 62$ with at most $\frac{u^2}{50}-\frac{11u}{100}-\frac{116}{75}$ triples has an embedding of order $v$ for each admissible integer $v \geq \frac{8u+17}{5}$., Comment: 20 pages, 0 figures

A weak $c$-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with $c$ colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be weakly $c$-chromatic if $c$ is the smallest number of colours with which the design can be weakly coloured. In this paper we show that for all $c \geq 2$ and $k \geq 3$ with $(c,k) \neq (2,3)$, the obvious necessary conditions for the existence of a $(v,k,\lambda)$-BIBD are asymptotically sufficient for the existence of a weakly $c$-chromatic $(v,k,\lambda)$-BIBD., Comment: 26 pages, 0 figures

We show that the complete graph on $n$ vertices can be decomposed into $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is odd, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\cdots+m_t=\binom n2$. We also show that the complete graph on $n$ vertices can be decomposed into a perfect matching and $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is even, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\ldots+m_t=\binom n2-\frac n2$., Comment: 182 pages, 0 figures, A condensed version of this paper was published as "Cycle decompositions V: Complete graphs into cycles of arbitrary lengths" (see reference [24]). Here, we include supplementary data and some proofs which were omitted from that paper

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., Comment: 30 pages, no figures