Cycle factorizations of cycle products

AbstractLet n and k1,k2,…,kn be integers with n > 1 and ki ⩾ 2 for 1 ⩽ i ⩽ n. We show that there exists a Cs-factorization of Πi=1n C2ki if and only if s = 2t with 2 ⩽ t ⩽ k1 + ··· + kn. We also settle the problem of cycle factorizations of the d-cube

On partitions of finite vector spaces of low dimension over GF(2)

AbstractLet Vn(q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of Vn(q) is a partition of Vn(q) if every nonzero vector in Vn(q) is contained in exactly one subspace of P. If there exists a partition of Vn(q) containing ai subspaces of dimension ni for 1≤i≤k, then (ak,ak−1,…,a1) must satisfy the Diophantine equation ∑i=1kai(qni−1)=qn−1. In general, however, not every solution of this Diophantine equation corresponds to a partition of Vn(q). In this article, we determine all solutions of the Diophantine equation for which there is a corresponding partition of Vn(2) for n≤7 and provide a construction of each of the partitions that exist

Mathematics: Models and Applications

xiii, 525 hlm.; Ind.; 26 c

Partitions of finite vector spaces over GF(2) into subspaces of dimensions 2 and s

AbstractA vector space partition of a finite vector space V over the field of q elements is a collection of subspaces whose union is all of V and whose pairwise intersections are trivial. While a number of necessary conditions have been proved for certain types of vector space partitions to exist, the problem of the existence of partitions meeting these conditions is still open. In this note, we consider vector space partitions of a finite vector space over the field GF(2) into subspaces of dimensions 2 and s.While certain cases have been done previously (s=1, s=3, and s even), in our main theorem we build upon these general results to give a constructive proof for the existence of vector space partitions over GF(2) into subspaces of dimensions s and 2 of almost all types. In doing so, we introduce techniques that identify subsets of our vector space which can be viewed as the union of subspaces having trivial pairwise intersection in more than one way. These subsets are used to transform a given partition into another partition of a different type. This technique will also be useful in constructing further partitions of finite vector spaces

Factorizations of and by powers of complete graphs

Let K-k(d) denote the Cartesian product of d copies of the complete graph K-k. We prove necessary and sufficient conditions for the existence of a K-k(r)-factorization of K-pn(s), where p is prime and k > 1, n, r and s are positive integers. (C) 2002 Elsevier Science B.V. All rights reserved

Labelings of unions of up to four uniform cycles

We show that every 2-regular graph consisting of at most four uniform components has a ρ-labeling (or a more restricted labeling). This has an application in the cyclic decomposition of certain complete graphs into the disjoint unions of cycles