John P. McSorley

Totally Magic Graphs

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.

Double Arrays, Triple Arrays and Balanced Grids

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.

Cyclic Permutations in Doubly-Transitive Groups

INTRODUCTION Let Ω be a finite set of size n. A cyclic permutation on Ω is a permutation whose cycle decomposition is one cycle of length n. This paper classifies all finite doubly-transitive permutation groups which contain a cyclic permutation. The classification appears in Table 1. We use (G, Ω) for a finite doubly-transitive permutation group G acting on a finite set Ω. For other notation and definitions see the self-contained article Cameron [1].

Counting structures in the Mo¨bius ladder

Abstract The Mobius ladder, M n , is a simple cubic graph on 2 n vertices. We present a technique which enables us to count exactly many different structures of M n , and somewhat unifies counting in M n . We also provide new combinatorial interpretations of some sequences, and ask some questions concerning extremal properties of cubic graphs.

Double Arrays, Triple Arrays and Balanced Grids with v=r+c -1

In Theorem 6.1 of McSorley et al. [3] it was shown that, when v=r+c−1, every triple array TA(v,k,λrr,λcc,k:r× c) is a balanced grid BG(v,k,k:r × c). Here we prove the converse of this Theorem. Our final result is: Let v=r+c−1. Then every triple array is a TA(v,k,c−k,r−k,k:r× c) and every balanced grid is a BG(v,k,k:r× c), and they are equivalent.

Single-change circular covering designs

Abstract A single-change circular covering design (scccd) based on the set [ v ] = {1,…, v } with block size k is an ordered collection of b blocks, B = {B 1 ,…,B b } , each B i ⊂ [ v ], which obey: (1) each block differs from the previous block by a single element, as does the last from the first, and, (2) every pair of [ v ] is covered by some block. The object is to minimize b for a fixed r and k . We present some minimal constructions of scccds for arbitrary v when k − 2 and 3, and for arbitrary k when k + 1 ⩽ v ⩽ 2 k . Tight designs are those in which each pair is covered exactly once. Start-Finish arrays are used to construct tight designs when v > 2 k ; there are 2 non-isomorphic tight designs with ( v , k ) = (9, 4), and 12 with ( v , k ) = (10, 4). Some non-existence results for tight designs, and standardized, element-regular, perfect, and column-regular designs are considered.

Orthogonal arrays of strength three from regular 3-wise balanced designs

The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction.

Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

Starting with the zero-square “zeon algebra,” the connection with permanents is shown. Permanents of submatrices of a linear combination of the identity matrix and all-ones matrix lead to moment polynomials with respect to the exponential distribution. A permanent trace formula analogous to MacMahons master theorem is presented and applied. Connections with permutation groups acting on sets and the Johnson association scheme arise. The families of numbers appearing as matrix entries turn out to be related to interesting variations on derangements. These generalized derangements are considered in detail as an illustration of the theory.

Combinatorial Interpretation and Operator Calculus of Lommel Polynomials

In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis.

Rhombic tilings of (n,k)-Ovals, (n,k,λ)-cyclic difference sets, and related topics

Abstract Each fixed integer n has associated with it ⌊ n 2 ⌋ rhombs: ρ 1 , ρ 2 , … , ρ ⌊ n 2 ⌋ , where, for each 1 ≤ h ≤ ⌊ n 2 ⌋ , rhomb ρ h is a parallelogram with all sides of unit length and with smaller face angle equal to h × π n radians. An Oval is a centro-symmetric convex polygon all of whose sides are of unit length, and each of whose turning angles equals l × π n for some positive integer l . A ( n , k ) -Oval is an Oval with 2 k sides tiled with rhombs ρ 1 , ρ 2 , … , ρ ⌊ n 2 ⌋ ; it is defined by its Turning Angle Index Sequence, a k -composition of n . For any fixed pair ( n , k ) we count and generate all ( n , k ) -Ovals up to translations and rotations, and, using multipliers, we count and generate all ( n , k ) -Ovals up to congruency. For odd n if a ( n , k ) -Oval contains a fixed number λ of each type of rhomb ρ 1 , ρ 2 , … , ρ ⌊ n 2 ⌋ then it is called a magic ( n , k , λ ) -Oval. We prove that a magic ( n , k , λ ) -Oval is equivalent to a ( n , k , λ ) -Cyclic Difference Set. For even n we prove a similar result. Using tables of Cyclic Difference Sets we find all magic ( n , k , λ ) -Ovals up to congruency for n ≤ 40 . Many related topics including lists of ( n , k ) -Ovals, partitions of the regular 2 n -gon into Ovals, Cyclic Difference Families, partitions of triangle numbers, u -equivalence of ( n , k ) -Ovals, etc., are also considered.