Walter D. Wallis

Edge-Magic Total Labelings

An edge-magic total labeling on a graph G is a one-to-one map λ from \( V (G) \ cup E(G) \) onto the integers 1,2, …, v + e, where v = | V (G) | and e = | E(G) |, with the property that, given any edge xy,

Clique partitions of the cocktail party graphs

\lambda (x) + \lambda (xy) + \lambda (y) = k

Kirkman's school projects

for some constant k. In other words, wt(xy) = k for any choice of edge xy. Then k is called the magic sum of G. Any graph with an edge-magic total labeling will be called edge-magic. described.

Vertex-magic labeling of trees and forests

A total labeling of a graph with v vertices and e edges is defined as a one-to-one map taking the vertices and edges onto the integers 1,2,···,v+e. Such a labeling is vertex magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and edge magic if the sum of an edge label and the labels of the endpoints of the edge is constant. In this paper we examine graphs possessing a labeling that is simultaneously vertex magic and edge magic. Such graphs appear to be rare.

Maximal sets of mutually orthogonal Latin squares

Abstract Let T v denote the complenent of a perfect matching in the complete graph on v vertices, v even, and let cp (T v ) be the minimum number of cliques needed to partition the edge-set of T v . We prove that cp (T v ) ⩾ v for v ⩾ 8 and give a design characterization of the cases where equality holds. We also show that, asymptotically, cp (T v ) ⩽ v log log v .

Skew-Hadamard Matrices and the Smith Normal Form

Abstract.Triple arrays are a class of designs introduced by Agrawal in 1966 for two-way elimination of heterogeneity in experiments. In this paper we investigate their existence and their connection to other classes of designs, including balanced incomplete block designs and balanced grids.

Vertex-Magic Total Labelings

Abstract We discuss the generalization of Kirkmans Schoolgirl Problem to the case where the number of schoolgirls is not a multiple of 3. It is required that all blocks be of size 3 except that there may be one block per round of size 2, or one of size 4.

Single change covering designs

A vertex-magic total labeling of a graph G(V, E) is a one-to-one map λ from E ∪ V onto the integers {1, 2,.....,|E| + |V|} such that λ(x) + Σ λ(xy), where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite different from the conjectured behavior of edge-magic total labelings of these graphs. We pay special attention to the case of so-called galaxies, forests in which every component tree is a star.