8 research outputs found
Cycle systems in the complete bipartite graph minus a one-factor
AbstractLet Kn,n−I denote the complete bipartite graph with n vertices in each part from which a 1-factor I has been removed. An m-cycle system of Kn,n−I is a collection of m-cycles whose edges partition Kn,n−I. Necessary conditions for the existence of such an m-cycle system are that m⩾4 is even, n⩾3 is odd, m⩽2n, and m|n(n−1). In this paper, we show these necessary conditions are sufficient except possibly in the case that m≡0(mod4) with n<m<2n
Decomposition of complete tripartite graphs into cycles and paths of length three
Let and denote a cycle and a path on vertices, respectively. In this paper, we obtain necessary and sufficient conditions for the decomposition of into copies of and copies of for all possible values of ,
Decomposing complete equipartite graphs into short odd cycles
In this paper we examine the problem of decomposing the lexicographic product of a cycle with an empty graph into cycles of uniform length. We determine necessary and sufficient conditions for a solution to this problem when the cycles are of odd length. We apply this result to find necessary and sufficient conditions to decompose a complete equipartite graph into cycles of uniform length, in the case that the length is both odd and shot relative to the number of parts
Cyclic cycle systems of the complete multipartite graph
In this paper, we study the existence problem for cyclic -cycle
decompositions of the graph , the complete multipartite graph with
parts of size , and give necessary and sufficient conditions for their
existence in the case that
Multipartite graph decomposition: cycles and closed trails
This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not K_n ), in the case that the cycle lengths are “small”. Cycles up to length n are considered, when the complete multipartite graph has n parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned
Path and cycle decompositions of complete equipartite graphs: 3 and 5 parts
In 1998 Cavenagh [N.J. Cavenagh, Decompositions of complete tripartite graphs into k-cycles, Australas. J. Combin. 18 (1998) 193-200] gave necessary and sufficient conditions for the existence of an edge-disjoint decomposition of a complete equipartite graph with three parts, into cycles of some fixed length k. Here we extend this to paths, and show that such a complete equipartite graph with three partite sets of size m, has an edge-disjoint decomposition into paths of length k if and only if k divides 3m(2) and k = 3 if and only if k divides 10m(2) and