The Hamiltonicity of bipartite graphs involving neighborhood unions (Q1598839)

Let \(G\) be a 2-connected graph with the partite sets \(X\) and \(Y\) such that \(|X|=|Y|=n\). The authors prove that if \(|N(x_1)\cup N(x_2)|+|N(y_1)\cup N(y_2)|\geq n+2\) for any pairs of vertices \(\{x_1,x_2\}\subseteq X\) and \(\{y_1,y_2\}\subseteq Y\), then \(G\) is Hamiltonian, unless \(G\) is a special graph of order 8 or of order 12.