Geodesic
Bounds for the number of meeting edges in graph partitioning
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 741-752 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $G$ be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that $G$ admits a bipartition such that each vertex class meets edges of total weight at least $(w_1-\Delta _1)/2+2w_2/3$, where $w_i$ is the total weight of edges of size $i$ and $\Delta _1$ is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph $G$ (i.e., multi-hypergraph), we show that there exists a bipartition of $G$ such that each vertex class meets edges of total weight at least $(w_0-1)/6+(w_1-\Delta _1)/3+2w_2/3$, where $w_0$ is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with $m$ edges, except for $K_2$ and $K_{1,3}$, admits a tripartition such that each vertex class meets at least $\lceil {2m}/{5}\rceil $ edges, which establishes a special case of a more general conjecture of Bollobás and Scott.
Let $G$ be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that $G$ admits a bipartition such that each vertex class meets edges of total weight at least $(w_1-\Delta _1)/2+2w_2/3$, where $w_i$ is the total weight of edges of size $i$ and $\Delta _1$ is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph $G$ (i.e., multi-hypergraph), we show that there exists a bipartition of $G$ such that each vertex class meets edges of total weight at least $(w_0-1)/6+(w_1-\Delta _1)/3+2w_2/3$, where $w_0$ is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with $m$ edges, except for $K_2$ and $K_{1,3}$, admits a tripartition such that each vertex class meets at least $\lceil {2m}/{5}\rceil $ edges, which establishes a special case of a more general conjecture of Bollobás and Scott.
DOI : 10.21136/CMJ.2017.0147-16
Classification : 05C35, 05C75
Keywords: graph; weighted hypergraph; partition; judicious partition 
@article{10_21136_CMJ_2017_0147_16,
     author = {Zeng, Qinghou and Hou, Jianfeng},
     title = {Bounds for the number of meeting edges in graph partitioning},
     journal = {Czechoslovak Mathematical Journal},
     pages = {741--752},
     year = {2017},
     volume = {67},
     number = {3},
     doi = {10.21136/CMJ.2017.0147-16},
     mrnumber = {3697913},
     zbl = {06770127},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0147-16/}
}
TY  - JOUR
AU  - Zeng, Qinghou
AU  - Hou, Jianfeng
TI  - Bounds for the number of meeting edges in graph partitioning
JO  - Czechoslovak Mathematical Journal
PY  - 2017
SP  - 741
EP  - 752
VL  - 67
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0147-16/
DO  - 10.21136/CMJ.2017.0147-16
LA  - en
ID  - 10_21136_CMJ_2017_0147_16
ER  - 
%0 Journal Article
%A Zeng, Qinghou
%A Hou, Jianfeng
%T Bounds for the number of meeting edges in graph partitioning
%J Czechoslovak Mathematical Journal
%D 2017
%P 741-752
%V 67
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0147-16/
%R 10.21136/CMJ.2017.0147-16
%G en
%F 10_21136_CMJ_2017_0147_16
Zeng, Qinghou; Hou, Jianfeng. Bounds for the number of meeting edges in graph partitioning. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 741-752. doi: 10.21136/CMJ.2017.0147-16

Cité par Sources :