Anthony J. W. Hilton

Hamiltonian decompositions of complete graphs

Abstract It is well known that K 2 n + 1 can be decomposed into n edge-disjoint Hamilton cycles. A novel method for constructing Hamiltonian decompositions of K 2 n + 1 is given and a procedure for obtaining all Hamiltonian decompositions of of K 2 n + 1 is outlined. This method is applied to find a necessary and sufficient condition for a decomposition of the edge set of K r ( r ≤ 2 n ) into n classes, each class consisting of disjoint paths to be extendible to a Hamiltonian decomposition of K 2 n + 1 so that each of the classes forms part of a Hamilton cycle.

Outlines of Latin Squares

A unified treatment of outline latin squares and of (p,q,r)-latin rectangles is presented, together with applications to embedding partial latin squares.

The chromatic index of a graph whose core has maximum degree two

Abstract The core G Δ of a simple graph G is the subgraph induced by the vertices of degree Δ = Δ( G ). In this paper we show that if G Δ has maximum degree two and G is Class 2, then G is critical. We also obtain a number of related results.

Amalgamations of connected k -factorizations

In this paper, necessary and sufficient conditions are found for a graph with exactly one amalgamated vertex to be the amalgamation of a k-factorization of Kkn+1 in which each k factor is connected. From this result, necessary and sufficient conditions for a given edge-coloured Kt to be embedded in a connected k-factorization of Kkn+1 are deduced.

Hamiltonian double Latin squares

A double latin square of order 2n on symbols σ1, ..., σn an is a 2n × 2n matrix A = (aij) in which each aij is one of the symbols σ1, ..., σn and each σk occurs twice in each row and twice in each column. For k = 1,...,n let B(A, σk) be the bipartite graph with vertices ρ1,..., ρ2n, c1, ..., c2n and 4n edges [ρi, cj] corresponding to ordered pairs (i,j) such that aij = σk. We say that A is Hamiltonian if B(A, σk) is a cycle of length 4n for k = 1 ,..., n. Two double latin squares (aij), (aij) of order 2n on symbols σ1,...,σn are said to be orthogonal if for each ordered pair (σh, σk) of symbols there are four ordered pairs (i,j) such that aij = σh, aij = σk.We explore ways of constructing Hamiltonian double latin squares (HLS), symmetric HLS, sets of mutually orthogonal HLS and pairs of orthogonal symmetric HLS. We identify those arrays which can be obtained from HLS by amalgamating rows and amalgamating columns in a certain sense, and we prove a similar result concerning symmetric arrays obtainable in this way from symmetric HLS. These results can be proved either by using matroids or by a more elementary method, and we illustrate both approaches. From these results we deduce a characterisation of those matrices which are submatrices of HLS on n symbols, a similar result concerning symmetric submatrices of symmetric HLS and some related results. Much of our discussion uses graph-theoretic language, since HLS on n symbols are equivalent to decompositions of K2n,2n into Hamiltonian cycles and symmetric HLS on n symbols are equivalent to decompositions of K2n into Hamiltonian paths (and these are equivalent to decompositions of K2n+1 into Hamiltonian cycles).

A sufficient condition for a regular graph to be class 1

The core GΔ of a simple graph G is the subgraph induced by the vertices of maximum degree. It is well known that the Petersen graph is not 1-factorizable and has property that the core of the graph obtained from it by removing one vertex has maximum degree 2. In this paper, we prove the following result. Let G be a regular graph of even order with d(G) ≥ 3. Suppose that G contains a vertex ν such that the core of Gν has maximum degree 2. If G is not the Petersen graph, then G is 1-factorizable.

Fractional latin squares, simplex algebras, and generalized quotients

Abstract Recently the authors defined the concept of a weighted quasigroup, and showed that each weighted quasigroup is the amalgamation of a quasigroup. Similar results were obtained for symmetric and other types of quasigroups. Here we first introduce the closely related concept of a fractional latin square and show that every fractional latin square is the fractional amalgamation of a latin square; we also show that every symmetric fractional latin square is the fractional amalgamation of a symmetric latin square; and we obtain some further similar results about some other types of fractional latin squares. We then introduce the concept of a simplex algebra, and we show that our results about weighted quasigroups and fractional latin squares can be given a very natural geometric formulation in terms of projections of simplex algebras. We also show how our results can be viewed as statements concerning a generalization of the standard quotient construction defined on an algebraic system.

The total chromatic number of nearly complete bipartite graphs

The total chromatic number χT(G) of a graph G is the least number of colours needed to colour the edges and vertices of G so that no two adjacent vertices receive the same colour, no two edges incident with the same vertex receive the same colour, and no edge receives the same colour as either of the vertices it is incident with. Let n ≥ 1, let J be a subgraph of Kn,n, let e = |E(J)|, and let j(J) be the maximum size (i.e., number of edges) of a matching in J. Then nχT(Kn,nE(J)) = n + 2 nif and only if e + j ≤ n − 1.

Outline symmetric latin squares

We define two symmetrical analogues of the notion of an outline latin square; we prove that each is an amalgamation of a simple symmetrical latin square like structure; and we show that various embedding theorems are consequences of these results.

Graphs that admit 3-to-1 or 2-to-1 maps onto the circle

Abstract We give a complete and explicit characterization of the connected graphs which admit a continuous 3-to-l map onto the circle, and of the connected graphs which admit a continuous 2-to-1 map onto the circle. This generalizes earlier work of Heath and Hilton who considered the mappings of trees onto the circle.