C. Vanden Eynden

Factorizations of complete multipartite graphs into generalized cubes

A graph is t-tough if the number of components of G\S is at most |S|-t for every cutset S ⊆ V (G). A k-walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1-walk is a Hamilton cycle, and a longstanding conjecture by Chvatal is that every sufficiently tough graph has a 1-walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficiently tough graph has a k-walk. We fill in the gap between k = 1 and k ≥ 3 by showing that, when k = 2, every sufficiently tough (specifically, 4-tough) graph has a 2-walk. To do this we first provide a new proof for and generalize a result by Win on the existence of a k-tree, a spanning tree with every vertex of degree at most k. We also provide new examples of tough graphs with no k-walk for k ≥ 2.

On a generalization of the Oberwolfach problem

Let e1, e2, ..., en be a sequence of nonnegative integers such that the first non-zero term is not one. Let Σi - 1n ei = (q - 1)/2, where q = pn and p is an odd prime. We prove that the complete graph on q vertices can be decomposed into e1 Cpn-factors, e2 Cpn - 1-factors, ..., and en Cp-factors.

Cycle factorizations of cycle products

Abstract Let n and k 1 , k 2 ,…, k n be integers with n > 1 and k i ⩾ 2 for 1 ⩽ i ⩽ n . We show that there exists a C s -factorization of Π i =1 n C 2 k i if and only if s = 2 t with 2 ⩽ t ⩽ k 1 + ··· + k n . We also settle the problem of cycle factorizations of the d -cube.

On λ-fold Rosa-type Labelings of Bipartite Multigraphs

Abstract It is known that for a given (simple) graph G with n edges, there exits a cyclic G-decomposition of K 2 n + 1 if and only if G admits a ρ-labeling. It is also known that if G is bipartite and it admits an ordered ρ-labeling, then there exists a cyclic G-decomposition of K 2 n x + 1 for every positive integer x. We extend these concepts to labelings of multigraphs through what we call λ-fold ρ-labelings and ordered λ-fold ρ-labelings. Let K m λ denote the λ-fold complete graph of order m. We sho that if a subgraph G of K 2 n / λ + 1 λ has size n, there exits a cyclic G-decomposition of K 2 n / λ + 1 λ if and only if G admits a λ-fold ρ-labeling. If in addition G is bipartite and it admits an ordered λ-fold ρ-labeling, then there exists a cyclic G-decomposition of K 2 n x / λ + 1 λ for every positive integer x. We discuss some classes of graphs and multigraphs that admit such labelings.

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

Abstract A 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.