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

Keywords

  • Bipartite graph
  • Degree
  • Longest cycle

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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