Hamiltonian cycles in bipartite graphs (Q1894704)

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.