Hamiltonian cycles in bipartite graphs
type: Item
instance of: scholarly article
title: Hamiltonian cycles in bipartite graphs
zbMATH Open document ID: 0832.05069
DOI: 10.1007/BF01200758
published in: Combinatorica
publication date: 1995-07-24T00:00:00Z
review text: Let \(G= (X, Y; E)\) be a balanced bipartite graph with vertex classes \(X\), \(Y\), edge set \(E\), and \(|X|= |Y|= n\). The balanced independence number \(\alpha^*(G)\) is defined to be \[ \max\{|A|: A\subseteq X\cup Y\wedge A\text{ is independent }\wedge \bigl||A\cap X|- |A\cap Y|\bigr|\leq 1\}. \] For \(A, B\subseteq X\cup Y\), let \([A, B]= \{ab: a\neq b\wedge a\in A\wedge b\in B\}\). The edge-density \(\lambda(G)\) is defined to be \[ \min\Biggl\{{|E\cap ([A, Y- B]\cup [X- A, B])|\over (|A|+ |B|)n- 2|A||B|}: A\subseteq X\wedge B\subseteq Y\wedge X\cup Y\neq A\cup B\neq \varnothing\Biggr\}. \] The following main result is proved: If \(\alpha^*(G)\leq n\lambda(G)- 1\) then \(G\) is Hamiltonian. The author remarks that using this result it can be shown that in the Hamiltonian game on the complete bipartite graph \(K_{n,n}\) the maker can achieve \({2\over 37}n\) edge-disjoint Hamiltonian cycles.
Mathematics Subject Classification ID: 05C45
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 778336
zbMATH Keywords: balanced bipartite graph
zbMATH Keywords: balanced independence number
zbMATH Keywords: edge-density
zbMATH Keywords: complete bipartite graph
zbMATH Keywords: Hamiltonian cycles
author: Xiaoyun Lu
reviewed by: Günter Schaar
MaRDI profile type: Publication
cites work: Remarks on positional games. I
cites work: Variations on a game
cites work: Van der Waerden and Ramsey type games
cites work: On a combinatorial game
cites work: Hamiltonian games
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description: scientific article; zbMATH DE number 778336