Daniel Horsley

Decompositions into 2-regular subgraphs and equitable partial cycle decompositions

Two theorems are proved in this paper. Firstly, it is proved that there exists a decomposition of the complete graph of order n into t edge-disjoint 2-regular subgraphs of orders m1, m2,...,mt if and only if n is odd, 3≤mi ≤ n for i = 1, 2,...,t, and m1 + m2 +...+ mt = (n 2). Secondly, it is proved that if there exists partial decomposition of the complete graph Kn of order n into t cycles of lengths m1, m2,..., mt, then there exists an equitable partial decomposition of Kn into t cycles of lengths m1, m2,..., mt. A decomposition into cycles is equitable if for any two vertices u and v, the number of cycles containing u and the number of cycles containing v differ by at most 1.

Cycle decompositions V: Complete graphs into cycles of arbitrary lengths

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 m1 + ··· + mt = n2- n/2.

Packing cycles in complete graphs

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.

Compressive Sensing Matrices and Hash Families

Deterministic construction of measurement matrices for compressive sensing can be effected by first constructing a relatively small matrix explicitly, and then inflating it using a column replacement technique to form a large measurement matrix that supports at least the same level of sparsity. In particular, using easily developed null space conditions for l0- and l1-recoverability, properties of the pattern matrix used to select columns lead to well-studied matrices, separating and distributing hash families. Two-stage compression and recovery techniques are developed that employ more computationally intensive l0-recoverability for small matrices and simpler l1-recoverability for one larger matrix; this can reduce the number of measurements required.

Decomposing Various Graphs into Short Even-Length Cycles

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.

Frameproof codes and compressive sensing

Fingerprinting of digital content is often employed to prevent a small coalition of legitimate users from constructing a copy whose fingerprint is registered to a user not in the coalition. Sets of fingerprints that prevent t or fewer users from framing another user in this way are frameproof codes. A frameproof code is termed secure when no fingerprint constructed by a coalition of t or fewer users can also be constructed by a disjoint coalition of t or fewer users. Secure frameproof codes are related to cover-free families arising in combinatorial group testing. Here a different connection is explored. Interpreting frameproof codes and secure frameproof codes as certain types of separating hash families, it is shown that each underlies a useful measurement matrix for compressive sensing. Indeed frameproof codes for coalitions of t users underlie measurement matrices that admit ℓ0-recoverability of t-sparse vectors, while secure frameproof codes for coalitions of t users underlie measurement matrices that admit ℓ1-recoverability of t-sparse vectors. Consequences for the construction of measurement matrices are briefly outlined, but the focus is on the combinatorial similarities of frameproof codes, separating and distributing hash families, and measurement matrices.

Maximum packings of the complete graph with uniform length cycles

In this paper we find the maximum number of pairwise edge-disjoint m-cycles which exist in a complete graph with n vertices, for all values of n and m with 3≤m≤n.

Decompositions of complete multigraphs into cycles of varying lengths

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.

Strengthening hash families and compressive sensing

The deterministic construction of measurement matrices for compressive sensing is a challenging problem, for which a number of combinatorial techniques have been developed. One of them employs a widely used column replacement technique based on hash families. It is effective at producing larger measurement matrices from smaller ones, but it can only preserve the strength (level of sparsity supported), not increase it. Column replacement is extended here to produce measurement matrices with larger strength from ingredient arrays with smaller strength. To do this, a new type of hash family, called a strengthening hash family, is introduced. Using these hash families, column replacement is shown to increase strength under two standard notions of recoverability. Then techniques to construct strengthening hash families, both probabilistically and deterministically, are developed. Using a variant of the Stein-Lovasz-Johnson theorem, a deterministic, polynomial time algorithm for constructing a strengthening hash family of fixed strength is derived.

Generalising Fisher’s inequality to coverings and packings

In 1940 Fisher famously showed that if there exists a non-trivial (v,k,λ)-design, then λ(v-1)⩾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,λ)-coverings and -packings with λ(v-1)