Brian Alspach - Academia.edu (original) (raw)

Papers by Brian Alspach

arXiv (Cornell University), Jul 9, 2020

Australas. J Comb., 2017

We complete the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian ... more We complete the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups.

arXiv (Cornell University), Jul 24, 2019

A partial Steiner triple system of order n is sequenceable if there is a sequence of length n of ... more A partial Steiner triple system of order n is sequenceable if there is a sequence of length n of its distinct points such that no proper segment of the sequence is a union of point-disjoint blocks. We prove that if a partial Steiner triple system has at most three point-disjoint blocks, then it is sequenceable.

Canadian Journal of Mathematics, Apr 1, 1982

arXiv (Cornell University), Nov 21, 2014

Various results on factorisations of complete graphs into circulant graphs and on 2factorisations... more Various results on factorisations of complete graphs into circulant graphs and on 2factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2p where p ≡ 5 (mod 8) is prime.

Canadian mathematical bulletin, Sep 1, 1970

arXiv (Cornell University), Jul 10, 2020

Honeycomb toroidal graphs are trivalent Cayley graphs on generalized dihedral groups. We examine ... more Honeycomb toroidal graphs are trivalent Cayley graphs on generalized dihedral groups. We examine the two historical threads leading to these graphs, some of the properties that have been established, and some open problems.

Journal of Graph Theory, Nov 5, 2012

We prove that connected Cayley graphs of valency at least 3 on abelian groups are even edge-pancy... more We prove that connected Cayley graphs of valency at least 3 on abelian groups are even edge-pancyclic and have cycles of every possible odd length bigger than or equal to the odd girth.

Discrete Mathematics, 2019

Abstract We classify trivalent vertex-transitive graphs whose edge sets have a partition into a H... more Abstract We classify trivalent vertex-transitive graphs whose edge sets have a partition into a Hamilton cycle and a 1-factor that is invariant under the action of the full automorphism group.

It has long been known that there exists a perfect magic cube of order n where n £ 3, 59 75 2m, a... more It has long been known that there exists a perfect magic cube of order n where n £ 3, 59 75 2m, and km with m odd and m >_ 1. That they do not exist for orders 2, 3, and 4 is not difficult to show. Recently, several authors have constructed perfect magic cubes of order 7. We shall give a method for constructing perfect magic cubes of orders n km with m odd and m _> 7. ?. INTRODUCTION A magic square of order n is an nx n arrangement of the integers 1,2, . „., n so that the sum of the integers in every row, column and the two main diagonals is n(n+ 1)12% the magic sum. Magic squares of orders 5 and 6 are shown in Figure 1. 20 9 23 12 1 22 11 5 19 8 4 18 7 21 15 6 25 14 3 17 13 2 16 10 24 1 29 30 6 10 35 34 11 22 17 24 3 33 18 23 12 21 4 32 20 13 26 15 5 9 25 16 19 14 28 2 8 7 31 27 36 It is a well-known and long established fact that there exists a magic square of every order n, n ^ 2. For details of these constructions, the reader is referred to W. S. Andrews [2], Maurice Krait...

Ars Mathematica Contemporanea, 2022

We examine the chromatic index of generalized truncations of graphs and multigraphs.

50 Years of Combinatorics, Graph Theory, and Computing, 2019

Bull. ICA, 2021

Australas. J Comb., 2021

A broader definition of generalized truncations of graphs is introduced followed by an exploratio... more A broader definition of generalized truncations of graphs is introduced followed by an exploration of some standard concepts and parameters with regard to generalized truncations.

Let Sc{l,...,n-1) satisfy -S =S mod n. The circulant graph G(n, S) with vertex set IQ, IJ l,“‘, u... more Let Sc{l,...,n-1) satisfy -S =S mod n. The circulant graph G(n, S) with vertex set IQ, IJ l,“‘, u,_r} and edge set E satisfies “iiIj E E if and only if i - i E S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = -S. Adarn conjectured that G(n, S)=G(n, S’) if and only if S = US’ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p2, and determine exact conditions on S that it be tme in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime, or n is divisible by 24.

First I want to say a few words about my graph terminology. If I want to allow loops, I use the a... more First I want to say a few words about my graph terminology. If I want to allow loops, I use the adjective reflexive. If I want to allow multiple edges, I use multigraph. Thus, a graph has no loops and no multiple edges. I use valency rather than degree. If we say a graph is 4-valent (or tetravalent), it means it is regular of valency 4, for example. A decomposition of a graph X is a partition of its edge set into subgraphs. There are two typical situations. Either we want all the subgraphs to be isomorphic to some fixed graph Y . We shall call this a Y -decomposition of X, or a decomposition of X into subgraphs isomorphic to Y . The other typical situation is that we are given a list L of subgraphs and we want a 1-1 correspondence between the parts of the decomposition and the members of L.

The Art of Discrete and Applied Mathematics, 2020

A group is strongly sequenceable if every connected Cayley digraph on the group admits an orthogo... more A group is strongly sequenceable if every connected Cayley digraph on the group admits an orthogonal directed cycle or an orthogonal directed path. This paper deals with the problem of whether finite abelian groups are strongly sequenceable. A method based on posets is used to show that if the connection set for a Cayley digraph on an abelian group has cardinality at most nine, then the digraph admits either an orthogonal directed path or an orthogonal directed cycle.