We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m; where 2pmpn: r 2003 Elsevier Science (USA). All rights reserved.

In this paper, we prove that directed cyclic hamiltonian cycle systems of the complete symmetric digraph, K∗ n, exist if and only if n ≡ 2 (mod 4) and n 6= 2p with p prime and α ≥ 1. We also show that directed cyclic hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (Kn− I)∗, exist if and only if n ≡ 0 (mod 4).

Let G be a vertex-disjoint union of directed cycles in the complete directed graph Dt , let |E(G)| be the number of directed edges of G and suppose G 6= ⇀ C 2 ∪ ⇀ C 3 or ⇀ C 5 if t = 5, and G 6= ⇀ C 3 ∪ ⇀ C 3 if t = 6. It is proved in this paper that for each positive integer t , there exist ⇀ C 3-decompositions for Dt − G if and only if t(t − 1)− |E(G)| ≡ 0 (mod 3). © 2008 Elsevier B.V. All ri...

Caccetta and Haggkvist 111 conjectured that every directed graph with n vertices and minimum outdegree k contains a (directed) cycle of length at most [n/k\. (T o see that this bound cannot be improved, consider the directed graph with n cyclically arranged vertices and with edges leading from each vertex u to the first k vertices following u in the cyclic order.) Our main result (Theorem 2) is...

Self-directed learning skills are known to influence successful outcomes in a higher education environment. Moreover, first year students entering lack self-directed skills, as these not always developed at school level. A digital promise tool was used for first-year student cohort reflect on their personal experiences and behaviour over period of two semesters. The researchers the augmented co...

Jirasek, J., Some remarks on Adam’s conjecture for simple directed graphs, Discrete Mathematics 108 (1992) 327-332. The classes of multidigraphs for which Adam’s conjecture (that any digraph containing a directed cycle has an arc whose reversal decreases the total number of directed cycles) does not hold, were described in Grinberg (1988), Jirasek (1987) and Thomassen (1987). The question remai...

We present a method to detect simple cycles of length 4 of a directed graph in O n1 ωe2 2 ω steps, where n denotes the number of nodes, e denotes the number of edges and ω is the exponent of matrix multiplication. This improves upon the currently fastest methods for α 2 4 ω ω 1 2 , where e nα.

Team Tennis competitions produce aggregate scores for teams, and thus team rankings, based on head-to-head matchups of individual team members. Similar scoring rules can be used to rank any two groups that must be compared on the basis of paired elements. We explore such rules in terms of their strategic and social choice characteristics, with particular emphasis on the role of cycles. We first...

We consider the problem of computing a minimum cycle basis in a directed graph. The input to this problem is a directed graph G whose edges have non-negative weights. A cycle in this graph is actually a cycle in the underlying undirected graph with edges traversable in both directions. A {−1,0,1} edge incidence vector is associated with each cycle: edges traversed by the cycle in the right dire...

We identify the class of directed one-trees and prove the so-called min-max theorem for them. As a consequence, we establish the equality of directed tree-width and a new measure, d-width, on this class of graphs. In addition, we prove a property of all directed one-trees and use this property to create an O(n) recognition algorithm and an O(n) algorithm for solving the Hamiltonian cycle proble...