On the longest path in a recursively partitionable graph
A connected graph \(G\) with order \(n \geq 1\) is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to \(K_1\), or for every sequence \((n_1, \ldots , n_p)\) of positive integers summing up to \(n\) there exists a partition \((V_1, \ldots , V_p)\) of \(V(G...
Main Author: | |
---|---|
Format: | Article |
Language: | English |
Published: |
AGH Univeristy of Science and Technology Press
2013-01-01
|
Series: | Opuscula Mathematica |
Subjects: | |
Online Access: | http://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3334.pdf |
id |
doaj-4c6a0c8e796a4c12af2f6dcc67b18343 |
---|---|
record_format |
Article |
spelling |
doaj-4c6a0c8e796a4c12af2f6dcc67b183432020-11-24T20:49:56ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742013-01-01334631640http://dx.doi.org/10.7494/OpMath.2013.33.4.6313334On the longest path in a recursively partitionable graphJulien Bensmail0Univ. Bordeaux LaBRI, UMR 5800, F-33400 Talence, FranceA connected graph \(G\) with order \(n \geq 1\) is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to \(K_1\), or for every sequence \((n_1, \ldots , n_p)\) of positive integers summing up to \(n\) there exists a partition \((V_1, \ldots , V_p)\) of \(V(G)\) such that each \(V_i\) induces a connected R-AP subgraph of \(G\) on \(n_i\) vertices. Since previous investigations, it is believed that a R-AP graph should be 'almost traceable' somehow. We first show that the longest path of a R-AP graph on \(n\) vertices is not constantly lower than \(n\) for every \(n\). This is done by exhibiting a graph family \(\mathcal{C}\) such that, for every positive constant \(c \geq 1\), there is a R-AP graph in \(\mathcal{C}\) that has arbitrary order \(n\) and whose longest path has order \(n-c\). We then investigate the largest positive constant \(c' \lt 1\) such that every R-AP graph on \(n\) vertices has its longest path passing through \(n \cdot c'\) vertices. In particular, we show that \(c' \leq \frac{2}{3}\). This result holds for R-AP graphs with arbitrary connectivity.http://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3334.pdfrecursively partitionable graphlongest path |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Julien Bensmail |
spellingShingle |
Julien Bensmail On the longest path in a recursively partitionable graph Opuscula Mathematica recursively partitionable graph longest path |
author_facet |
Julien Bensmail |
author_sort |
Julien Bensmail |
title |
On the longest path in a recursively partitionable graph |
title_short |
On the longest path in a recursively partitionable graph |
title_full |
On the longest path in a recursively partitionable graph |
title_fullStr |
On the longest path in a recursively partitionable graph |
title_full_unstemmed |
On the longest path in a recursively partitionable graph |
title_sort |
on the longest path in a recursively partitionable graph |
publisher |
AGH Univeristy of Science and Technology Press |
series |
Opuscula Mathematica |
issn |
1232-9274 |
publishDate |
2013-01-01 |
description |
A connected graph \(G\) with order \(n \geq 1\) is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to \(K_1\), or for every sequence \((n_1, \ldots , n_p)\) of positive integers summing up to \(n\) there exists a partition \((V_1, \ldots , V_p)\) of \(V(G)\) such that each \(V_i\) induces a connected R-AP subgraph of \(G\) on \(n_i\) vertices. Since previous investigations, it is believed that a R-AP graph should be 'almost traceable' somehow. We first show that the longest path of a R-AP graph on \(n\) vertices is not constantly lower than \(n\) for every \(n\). This is done by exhibiting a graph family \(\mathcal{C}\) such that, for every positive constant \(c \geq 1\), there is a R-AP graph in \(\mathcal{C}\) that has arbitrary order \(n\) and whose longest path has order \(n-c\). We then investigate the largest positive constant \(c' \lt 1\) such that every R-AP graph on \(n\) vertices has its longest path passing through \(n \cdot c'\) vertices. In particular, we show that \(c' \leq \frac{2}{3}\). This result holds for R-AP graphs with arbitrary connectivity. |
topic |
recursively partitionable graph longest path |
url |
http://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3334.pdf |
work_keys_str_mv |
AT julienbensmail onthelongestpathinarecursivelypartitionablegraph |
_version_ |
1716805333885648896 |