𝔖 Bobbio Scriptorium
✦   LIBER   ✦

The intersection problem for K4 - e designs

✍ Scribed by Elizabeth J. Billington; M. Gionfriddo; C.C. Lindner

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
886 KB
Volume
58
Category
Article
ISSN
0378-3758

No coin nor oath required. For personal study only.

✦ Synopsis

A G-design of order n is a pair (P,B) where P is the vertex set of the complete graph Kn and B is an edge-disjoint decomposition of Kn into copies of the simple graph G. Following design terminology, we call these copies "blocks". Here/Β£4 -e denotes the complete graph K4 with one edge removed. It is well-known that a K4 -e design of order n exists if and only if n ~ 0 or 1 (mod 5), n~>6. The intersection problem here asks for which k is it possible to find two K4 -e designs (P, BI) and (P, B2) of order n, with ]B1 NB2I = k, that is, with precisely k common blocks. Here we completely solve this intersection problem for K4 -e designs.

πŸ“œ SIMILAR VOLUMES

On the existence of resolvable K4 βˆ’ e de
On the existence of resolvable K4 βˆ’ e designs
✍ Gennian Ge; Alan C. H. Ling πŸ“‚ Article πŸ“… 2007 πŸ› John Wiley and Sons 🌐 English βš– 115 KB πŸ‘ 1 views

## Abstract In this article, we settle a problem which originated in 4 regarding the existence of resolvable (__K__~4~β€‰βˆ’β€‰__e__)‐design. We solve the problem with two possible exceptions. Β© 2007 Wiley Periodicals, Inc. J Combin Designs 15: 502–510, 2007

Minimum embedding of P3-designs into (K4
Minimum embedding of P3-designs into (K4β€”e)-designs
✍ Charles J. Colbourn; Alan C. H. Ling; Gaetano Quattrocchi πŸ“‚ Article πŸ“… 2003 πŸ› John Wiley and Sons 🌐 English βš– 173 KB πŸ‘ 1 views

## Abstract A (__K__~4~β€‰βˆ’β€‰__e__)‐design on __v__ + __w__ points __embeds__ a __P__~3~‐design on __v__ points if there is a subset of __v__ points on which the __K__~4~β€‰βˆ’β€‰__e__ blocks induce the blocks of a __P__~3~‐design. It is shown that __w__ β‰₯ ¾(__v__β€‰βˆ’β€‰1). When equality holds, the embedding de

The intersection problem for m-cycle sys
The intersection problem for m-cycle systems
✍ Elizabeth J. Billington πŸ“‚ Article πŸ“… 1993 πŸ› John Wiley and Sons 🌐 English βš– 794 KB

Let Z,(v) denote the set of integers k for which a pair of m-cycle systems of K , exist, on the same vertex set, having k common cycles. Let J,(v) = { 0,1,2, . . . ,t, -2, t,} where t , = v(vl ) / 2 m . In this article, if 2mn + x is an admissible order of an m-cycle system, we investigate when Zm(2