Anthony B. Evans

Representations of graphs and orthogonal latin square graphs

We define graph representations modulo integers and prove that any finite graph has a representation modulo some integer. We use this to obtain a new, simpler proof of Lindner, E. Mendelsohn, N. Mendelsohn, and Wolks result that any finite graph can be represented as an orthogonal latin square graph.

Latin Squares without Orthogonal Mates

In 1779 Euler proved that for every even n there exists a latin square of order n that has no orthogonal mate, and in 1944 Mann proved that for every n of the form 4k + 1, k ≥ 1, there exists a latin square of order n that has no orthogonal mate. Except for the two smallest cases, n = 3 and n = 7, it is not known whether a latin square of order n = 4k + 3 with no orthogonal mate exists or not. We complete the determination of all n for which there exists a mate-less latin square of order n by proving that, with the exception of n = 3, for all n = 4k + 3 there exists a latin square of order n with no orthogonal mate. We will also show how the methods used in this paper can be applied more generally by deriving several earlier non-orthogonality results.

On orthogonal orthomorphisms of cyclic and non-abelian groups

Abstract We prove that if m >3 is odd and not divisible by 9 then we can construct a pair of orthogonal orthomorphisms of Z m . From this we derive new lower bounds on the number of pairwise orthogonal orthomorphisms of classes of dihedral groups of doubly even order, and classes of linear groups.

Maximal sets of mutually orthogonal Latin squares. I

One problem of interest in the study of latin squares is that of determining parameter pairs ( n , r ) for which there exists a maximal set of r mutually orthogonal latin squares of order n . In this paper we find new such parameter pairs by constructing maximal sets of mutually orthogonal latin squares using difference matrices. In the process we generalize known non-existence results for complete mappings, strong complete mappings and Knut Vic designs.

Note: Representations of disjoint unions of complete graphs

In this paper we will present some new bounds on representation numbers and dimensions of disjoint unions of complete graphs. These representations are closely related to mutually orthogonal sets of latin squares.

Magic rectangles and modular magic rectangles

Abstract Magic rectangles are m × n matrices with entries 1, …, mn, all row sums being equal and all column sums being equal. Sun established necessary and sufficient conditions for the existence of magic (m, n) rectangles. We introduce modular magic rectangles, variants of magic rectangles, and study two classes of modular magic rectangles: Pseudomagic and complete magic rectangles. We construct classes of pseudomagic, modular magic rectangles that are not magic rectangles, and classes of complete, modular magic rectangles. This suggests the problem of determining the spectra of pseudomagic, modular magic rectangles that are not magic rectangles; complete, modular magic rectangles; and complete, magic rectangles.

Representation Numbers of Stars

Abstract A graph G has a representation modulo r if there exists an injective map ƒ : V(G) → {0, 1, . . . , r – 1} such that vertices u and 𝑣 are adjacent if and only if ƒ(u) – ƒ(𝑣) is relatively prime to r. The representation number rep(G) is the smallest positive integer r for which G has a representation modulo r. In this paper we study representation numbers of the stars K 1,n . We will show that the problem of determining rep(K 1,n ) is equivalent to determining the smallest even k for which φ(k) ≥ n: we will solve this problem for “small” n and determine the possible forms of rep(K 1,n ) for sufficiently large n.

Representation numbers of complete multipartite graphs

A graph G has a representation modulo r if there exists an injective map f:V(G)->{0,1,...,r-1} such that vertices u and v are adjacent if and only if f(u)-f(v) is relatively prime to r. The representation number rep(G) is the smallest r such that G has a representation modulo r. Following earlier work on stars, we study representation numbers of complete bipartite graphs and more generally complete multipartite graphs.

The existence of complete mappings of SL(2,q), q≡3 modulo 4

In 1955, Hall and Paige conjectured that any finite group with a noncyclic Sylow 2-subgroup admits complete mappings. For the groups GL(2,q), SL(2,q), PSL(2,q), and PGL(2,q) this conjecture has been proved except for SL(2,q), q=3 modulo 4. We prove the conjecture true for SL(2,q), q=3 modulo 4.

Generating orthomorphisms of GF( q )

Abstract A map θ : GF( q ) + →GF( q ) + , with O θ =0, is an orthomorphism of GF( q ) + if θ and the map η defined by xη = xθ − x are both bijections. Orthomorphisms can be used in the construction of sets of mutually orthogonal latin squares and in the construction of affine planes. In this paper we give a method for constructing classes of orthomorphisms of GF( q ) + . We also described all orthomorphisms of GF( q ) + for q ⩽ 8.