✦ LIBER ✦

On the construction of odd cycle systems

✍ Scribed by D. G. Hoffman; C. C. Lindner; C. A. Rodger

Publisher: John Wiley and Sons
Year: 1989
Tongue: English
Weight: 406 KB
Volume: 13
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.3190130405

No coin nor oath required. For personal study only.

✦ Synopsis

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 exists a 15-cycle system of order n if and only if n = 1, 15, 21, or 25 (mod 301, and there exists a 21-cycle system of order n if and only if n = 1, 7, 15, or 21 (mod 421, n.# 7, 15.

📜 SIMILAR VOLUMES

Rotational k-cycle systems of order v k;

Rotational k-cycle systems of order v k; another proof of the existence of odd cycle systems

✍ Marco Buratti 📂 Article 📅 2003 🏛 John Wiley and Sons 🌐 English ⚖ 112 KB

## Abstract We give an explicit solution to the existence problem for 1‐rotational __k__‐cycle systems of order __v__ < 3__k__ with __k__ odd and __v__ ≠ 2__k__ + 1. We also exhibit a 2‐rotational __k__‐cycle system of order 2__k__ + 1 for any odd __k__. Thus, for __k__ odd and any admissible __v__

On digraphs with the odd cycle property

✍ Rachel Manber; Jia-Yu Shao 📂 Article 📅 1986 🏛 John Wiley and Sons 🌐 English ⚖ 462 KB

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.

On hamiltonian cycles in the prism over

On hamiltonian cycles in the prism over the odd graphs

✍ Letícia R. Bueno; Peter Horák 📂 Article 📅 2010 🏛 John Wiley and Sons 🌐 English ⚖ 137 KB 👁 2 views

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

A bound on the chromatic number using th

A bound on the chromatic number using the longest odd cycle length

✍ Sreyash Kenkre; Sundar Vishwanathan 📂 Article 📅 2007 🏛 John Wiley and Sons 🌐 English ⚖ 120 KB 👁 1 views

## Abstract Let __G__ be a non‐bipartite graph with ℓ as the length of the longest odd cycle. Erdös and Hajnal proved that χ(__G__) ≤ ℓ + 1. We show that the only graphs for which this is tight are those that contain __K__~ℓ~ + 1 and further, if __G__ does not contain __K__~ℓ~ then χ(__G__) ≤ ℓ −1.

The multi-color Ramsey number of an odd

The multi-color Ramsey number of an odd cycle

✍ Yusheng Li 📂 Article 📅 2009 🏛 John Wiley and Sons 🌐 English ⚖ 73 KB

## Abstract Let __r__~__k__~(__G__) be the __k__‐color Ramsey number of a graph __G__. It is shown that $r\_{k}(C\_{5})\le \sqrt{18^{k}\,k!}$ for __k__⩾2 and that __r__~__k__~(__C__~2__m__+ 1~)⩽(__c__^__k__^__k__!)^1/__m__^ if the Ramsey graphs of __r__~__k__~(__C__~2__m__+ 1~) are not far away fr

On the construction of physical systems

On the construction of physical systems from spectral data

✍ Zimoch, R. Z. 📂 Article 📅 1990 🏛 Wiley (John Wiley & Sons) 🌐 English ⚖ 357 KB

## Abstract This paper presents an algorithm for the solution of the inverse problem for a matrix differential equation of the form Aẍ + Bẋ + Cx = 0. In the method use is made of the Taylor formula for a matrix function of many variables. It is shown that the algorithm is always convergent. The pap