Three obvious necessary conditions for the existence of a k-cycle system of order n are that if n > 1 then n 1 k, n is odd, and 2 k divides n(n -1). We show that if these necessary conditions are sufficient for all n satisfying k I n < 3k then they are sufficient for all n. In particular, there exis
On digraphs with the odd cycle property
β Scribed by Rachel Manber; Jia-Yu Shao
- Publisher
- John Wiley and Sons
- Year
- 1986
- Tongue
- English
- Weight
- 462 KB
- Volume
- 10
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
We say that a digraph D has the odd cycle property if there exists an edge subset S such that every cycle of D has an odd number of edges from S. We give necessary and sufficient conditions for a digraph to have the odd cycle property. We also consider the analogous problem for graphs.
π SIMILAR VOLUMES
We prove that Woodall's and GhouileHouri's conditions on degrees which ensure that a digraph is Hamiltonian, also ensure that it contains the analog of a directed Hamiltonian cycle but with one edge pointing the wrong way; that is, it contains two vertices that are connected in the same direction by
The Kneser graph K (n, k) has as its vertex set all k-subsets of an n-set and two k-subsets are adjacent if they are disjoint. The odd graph O k is a special case of Kneser graph when n = 2k +1. A long standing conjecture claims that O k is hamiltonian for all k>2. We show that the prism over O k is
## Abstract Let __G__ be a nonβbipartite graph with β as the length of the longest odd cycle. ErdΓΆs and Hajnal proved that Ο(__G__) β€ β + 1. We show that the only graphs for which this is tight are those that contain __K__~β~ + 1 and further, if __G__ does not contain __K__~β~ then Ο(__G__) β€ β β1.
A graph is constructed to provide a negative answer to the following question of Bondy: Does every diconnected orientation of a complete k-partite (k 2 5) graph with each part of size at least 2 yield a directed (k + 1)-cycle?
If C is a q-ary code of length n and a and b are two codewords, then c is called a descendant of a and b if c i # [a i , b i ] for i=1, ..., n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the ``parent'' codewords in C. We study