Long cycles in unbalanced bipartite graphs

Shuya Chiba, Jun Fujisawa, Masao Tsugaki, Tomoki Yamashita

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let G[X,Y] be a 2-connected bipartite graph with |X|<|Y|. For S⊆V(G), we define δ(S;G):=min dG(v):v∈S. We define σ1 ,1(G):=min dG(x)+ dG(y):x∈X, y∈Y,xy∉E(G) and σ2(X):=min dG(x)+ dG(y):x,y∈X,x≠y. We denote by c(G) the length of a longest cycle in G. Jackson [B. Jackson, Long cycles in bipartite graphs, J. Combin. Theory Ser. B 38 (1985) 118-131] proved that c(G)<min2δ(X;G) +2δ(Y;G)-2,4δ(X;G)-4,2|Y|. In this paper, we extend this result, and prove that c(G)<min2σ1 ,1(G)-2,2 σ2(X)-4, 2|Y|.

Original languageEnglish
Pages (from-to)1857-1862
Number of pages6
JournalDiscrete Mathematics
Volume312
Issue number11
DOIs
Publication statusPublished - 2012 Jun 6

Fingerprint

Long Cycle
Bipartite Graph
Connected graph
Denote

Keywords

  • Bipartite graph
  • Degree
  • Longest cycle

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Long cycles in unbalanced bipartite graphs. / Chiba, Shuya; Fujisawa, Jun; Tsugaki, Masao; Yamashita, Tomoki.

In: Discrete Mathematics, Vol. 312, No. 11, 06.06.2012, p. 1857-1862.

Research output: Contribution to journalArticle

Chiba, S, Fujisawa, J, Tsugaki, M & Yamashita, T 2012, 'Long cycles in unbalanced bipartite graphs', Discrete Mathematics, vol. 312, no. 11, pp. 1857-1862. https://doi.org/10.1016/j.disc.2012.02.019
Chiba, Shuya ; Fujisawa, Jun ; Tsugaki, Masao ; Yamashita, Tomoki. / Long cycles in unbalanced bipartite graphs. In: Discrete Mathematics. 2012 ; Vol. 312, No. 11. pp. 1857-1862.
@article{0e470cffbe7242aba551927227db359b,
title = "Long cycles in unbalanced bipartite graphs",
abstract = "Let G[X,Y] be a 2-connected bipartite graph with |X|<|Y|. For S⊆V(G), we define δ(S;G):=min dG(v):v∈S. We define σ1 ,1(G):=min dG(x)+ dG(y):x∈X, y∈Y,xy∉E(G) and σ2(X):=min dG(x)+ dG(y):x,y∈X,x≠y. We denote by c(G) the length of a longest cycle in G. Jackson [B. Jackson, Long cycles in bipartite graphs, J. Combin. Theory Ser. B 38 (1985) 118-131] proved that c(G),1(G)-2,2 σ2(X)-4, 2|Y|.",
keywords = "Bipartite graph, Degree, Longest cycle",
author = "Shuya Chiba and Jun Fujisawa and Masao Tsugaki and Tomoki Yamashita",
year = "2012",
month = "6",
day = "6",
doi = "10.1016/j.disc.2012.02.019",
language = "English",
volume = "312",
pages = "1857--1862",
journal = "Discrete Mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "11",

}

TY - JOUR

T1 - Long cycles in unbalanced bipartite graphs

AU - Chiba, Shuya

AU - Fujisawa, Jun

AU - Tsugaki, Masao

AU - Yamashita, Tomoki

PY - 2012/6/6

Y1 - 2012/6/6

N2 - Let G[X,Y] be a 2-connected bipartite graph with |X|<|Y|. For S⊆V(G), we define δ(S;G):=min dG(v):v∈S. We define σ1 ,1(G):=min dG(x)+ dG(y):x∈X, y∈Y,xy∉E(G) and σ2(X):=min dG(x)+ dG(y):x,y∈X,x≠y. We denote by c(G) the length of a longest cycle in G. Jackson [B. Jackson, Long cycles in bipartite graphs, J. Combin. Theory Ser. B 38 (1985) 118-131] proved that c(G),1(G)-2,2 σ2(X)-4, 2|Y|.

AB - Let G[X,Y] be a 2-connected bipartite graph with |X|<|Y|. For S⊆V(G), we define δ(S;G):=min dG(v):v∈S. We define σ1 ,1(G):=min dG(x)+ dG(y):x∈X, y∈Y,xy∉E(G) and σ2(X):=min dG(x)+ dG(y):x,y∈X,x≠y. We denote by c(G) the length of a longest cycle in G. Jackson [B. Jackson, Long cycles in bipartite graphs, J. Combin. Theory Ser. B 38 (1985) 118-131] proved that c(G),1(G)-2,2 σ2(X)-4, 2|Y|.

KW - Bipartite graph

KW - Degree

KW - Longest cycle

UR - http://www.scopus.com/inward/record.url?scp=84858206034&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858206034&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2012.02.019

DO - 10.1016/j.disc.2012.02.019

M3 - Article

VL - 312

SP - 1857

EP - 1862

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 11

ER -