Christopher A. Rodger

On the construction of odd cycle systems

Three obvious necessary conditions for the existence of a k-cycle system of order n are that if n > 1 then n ⩾ k, n is odd, and 2k divides n(n − 1). We show that if these necessary conditions are sufficient for all n satisfying k ⩽ n < 3k then they are sufficient for all n. In particular, there exists a 15-cycle system of order n if and only if n ≡ 1, 15, 21, or 25 (mod 30), and there exists a 21-cycle system of order n if and only if n ≡ 1, 7, 15, or 21 (mod 42), n ≠ 7. 15.

Connectivity of cages

A (k; g)-graph is a k-regular graph with girth g. Let f(k; g) be the smallest integer v such there exists a (k; g)-graph with v vertices. A (k; g)-cage is a (k; g)-graph with f(k; g) vertices. In this paper we prove that the cages are monotonic in that f(k; g1) < f(k; g2) for all k ≥ 3 and 3 ≥ g1 < g2. We use this to prove that (k; g)-cages are 2-connected, and if k = 3 then their connectivity is k.

Hamiltonian decompositions of complete regular s -partite graphs

Abstract In this paper we give a procedure by which Hamiltonian decompositions of the s-partite graph Kn,n,…,n, where (s − 1)n is even, can be constructed. For 2t⩽s, 1⩽a1⩽…⩽at⩽n, we find conditions which are necessary and sufficient for a decomposition of the edge-set of Ka1a2…,at into (s − 1)n/2 classes, each class consisting of disjoint paths, to be extendible to a Hamiltonian decomposition of the complete s-partite graph Krmn,n,…,n so that each of the classes forms part of a Hamiltonian cycle.

The chromatic index of complete multipartite graphs

We show that a complete multipartite graph is class one if and only if it is not eoverfull, thus determining its chromatic index.

Coding Theory and Cryptography: The Essentials

Coding theory: introduction to coding theory linear codes perfect and related codes cyclic linear codes BCH codes Reed-Solomon codes burst error-correcting codes convolutional codes Reed-Muller and Preparata codes. Cryptography: classicalcryptography topics in algebra and number theory public-key cryptography. Appendices: the Euclidean algorithm factorization of 1 + xn example of compact disc encoding solutions to selected exercises.

Constructions of perfect Mendelsohn designs

Abstract Let n and k be positive integers. An (n, k, 1)-Mendelsohn design is an ordered pair (V, C ) where V is the vertex set of Dn, the complete directed graph on n vertices, and C is a set of directed cycles (called blocks) of length k which form an arc-disjoint decomposition of Dn. An (n, k, 1)-Mendelsohn design is called a perfect design and denoted briefly by (n, k, 1)-PMD if for any r, 1⩽r⩽k-1, and for each (x, y) ϵ V×V there is exactly one cycle c∈ C in which the (directed) distance along c from x to y is r. A necessary condition for the existence of an (n, k, 1)-PMD is n(n-1)≡0(modk). In this paper we shall describe some new techniques used in the construction of PMDs, including constructions of the product type. As an application, we show that the necessary condition for the existence of an (n, 5, 1)-PMD is also sufficient, except for n=6 and with at most 21 possible exceptions of n of which 286 is the largest.

The spectrum for 2-perfect 6-cycle systems

Abstract Recently, the spectrum problem for 2-perfect m -cycle systems has been studied by several authors. In this paper we find the spectrum for 2-perfect 6-cycle systems with two possible exceptions. The connection between these systems and quasigroups satisfying some 2 variable identities is discussed.

Embeddings of m -cycle systems and incomplete m -cycle systems: m ≤14

In this paper we completely settle the embedding problem for m-cycle systems with m less than or equal to 14. We also solve the more general problem of finding m-cycle systems of K-v - K-u when m is an element of {4,6,7,8,10,12,14}.

The embedding of partial triple systems when 4 Divides l

Abstract We show that if 4 divides λ, then any partial triple system of order r and index λ can be embedded in a proper triple system of index λ and order n whenever n is λ-admissible and n ⩾ 2 r + 1. Moreover we find a set of necessary conditions for the embedding of a partial triple system of index λ when λ is even and show that when 4 divides λ, then a very closely related set of conditions is sufficient.

Class one graphs

Abstract Conditions on the subgraph induced by the vertices of maximum degree of a simple graph G are found which are sufficient for G to be class 1. These conditions can be used to generalize a result of Fournier. An efficient algorithm is described for deciding whether or not a particular graph satisfies the conditions.