C. A. Rodger

2-Perfect m -cycle systems

Abstract The spectrum for 2-perfect m -cycle systems of K n has been considered by several authors in the case when m ⩽7. In this paper we essentially solve the problem for 2-perfect m -cycle systems of K n in the case where m is prime and 2 m +1 is a prime power. In particular we settle the problem for m = 11 and 13 except for two or one possible exceptions respectively. The problem for m = 9 is also considered.

( k, g )-cages are 3-connected

Abstract A ( k, g )- cage is a graph that has the least number of vertices among all k -regular graphs with girth g . It has been conjectured that the connectivity of each ( k, g -cage is k , and a proof exists for k = 3. We prove here that all cages are 3-connected, a step towards a proof of the conjecture.

Packing complete multipartite graphs with 4-cycles

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge-disjoint 4-cycles, and the minimum leaves are explicitly given.

A partial m =(2 k +1)-cycle system of order n can be embedded in an m -cycle system of order (2 n +1) m

Abstract A generalization of Cruses Theorem on embedding partial idempotent commutative latin squares is developed and used to show that a partial m=(2k+1)-cycle system of order n can be embedded in an m-cycle system of order tm for every odd t⩾(2n+1).

Embedding an incomplete latin square in a latin square with a prescribed diagonal

Abstract We present necessary and sufficient conditions for the embedding of a given incomplete latin square R of side n,n⩾10 on the symbols σ1,…,σt in a latin square T of side t on the symbols σ1,…,σt for all t⩾2n+1 where the diagonal of T has been prescribed. The lower bound of 2n+1 for t is the best possible.

Hamilton decompositions of complete graphs with a 3-factor leave

Abstract We show that for any 2-factor U of K n with n even, there exists a 3-factor T of K n such that E ( U )⊂ E ( T ) such that K n − E ( T ) admits a hamilton decomposition. This is proved with the method of amalgamations (graph homomorphisms), using a new result that concerns graph decompositions that are fairly divided, but not necessarily regular.

Linear spaces with many small lines

Abstract In this paper some of the work in linear spaces in which most of the lines have few points is surveyed. This includes existence results, blocking sets and embeddings. Also, it is shown that any linear space of order v can be embedded in a linear space of order about 13v in which there are no lines of size 2.

On the number of edge-disjoint one factors and the existence of k -factors in complete multipartite graphs

Abstract In this paper we use Tuttes f -factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k -factor in the complete multipartite graph K ( p (1), …, p ( n )), conditions that are reminiscent of the Erdos-Gallai conditions for the existence of simple graphs with a given degree sequence. We then use this result to investigate the maximum number of edge-disjoint 1-factors in K ( p (1), …, p ( n )), settling the problem in the case where this number is greater than δ - p (2), where p (1) ⩽ p (2) ⩽…⩽ p ( n ).

Two Doyen-Wilson theorems for maximum packings with triples

In this paper we complete the work begun by Mendelsohn and Rosa and by Hartman, finding necessary and sufficient conditions for a maximum packing with triples of order m MPT(m) to be embedded in an MPT(n). We also characterize when it is possible to embed an MPT(m) with leave LI in an MPT(n) with leave L2 in such a way that L1 C L2.

Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames

A k -factor of a graph G = ( V ( G ) , E ( G ) ) is a k -regular spanning subgraph of G . A k -factorization is a partition of E ( G ) into k -factors. Let K ( n , p ) be the complete multipartite graph with p parts, each of size n . If V 1 , ? , V p are the p parts of V ( K ( n , p ) ) , then a holey ? k -factor of deficiency V i of K ( n , p ) is a k -factor of K ( n , p ) - V i for some i satisfying 1 ? i ? p . Hence a holey k -factorization is a set of holey k -factors whose edges partition E ( K ( n , p ) ) . A holey hamiltonian decomposition is a holey 2-factorization of K ( n , p ) where each holey 2-factor is a connected subgraph of K ( n , p ) - V i for some i satisfying 1 ? i ? p . A (holey) k -factorization of K ( n , p ) is said to be fair if the edges between each pair of parts are shared as evenly as possible among the permitted (holey) factors. In this paper the existence of fair holey hamiltonian decompositions of K ( n , p ) is completely settled. This result simultaneously settles the existence of cycle frames of type n p for cycles of the longest length, being a companion for results in the literature for frames with short cycle length.