19 research outputs found
Perfect 1-factorisations of circulants with small degree
A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed
Colorings of some Cayley graphs
Cayley graphs are graphs on algebraic structures, typically groups or
group-like structures. In this paper, we have obtained a few results on Cayley
graphs on Cyclic groups, typically powers of cycles, some colorings of powers
of cycles, Cayley graphs on some non-abelian groups, and Cayley graphs on
gyrogroups.Comment: 9 page
Sharply transitive 1-factorizations of complete multipartite graphs
Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic
Sharply transitive 1-factorizations of complete multipartite graphs
Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic
Invariant means on Boolean inverse monoids
The classical theory of invariant means, which plays an important role in the
theory of paradoxical decompositions, is based upon what are usually termed
`pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean
inverse monoids which give rise to etale topological groupoids under
non-commutative Stone duality. We accordingly initiate the theory of invariant
means on arbitrary Boolean inverse monoids. Our main theorem is a
characterization of when a Boolean inverse monoid admits an invariant mean.
This generalizes the classical Tarski alternative proved, for example, by de la
Harpe and Skandalis, but using different methods
A Pair of Disjoint 3-GDDs of type g^t u^1
Pairwise disjoint 3-GDDs can be used to construct some optimal
constant-weight codes. We study the existence of a pair of disjoint 3-GDDs of
type and establish that its necessary conditions are also sufficient.Comment: Designs, Codes and Cryptography (to appear