A spectral condition for Hamilton cycles in tough bipartite graphs

Let $G$ be a graph. The {\em spectral radius} of $G$ is the largest eigenvalue of its adjacency matrix. For a non-complete bipartite graph $G$ with parts $X$ and $Y$, the {\em bipartite toughness} of $G$ is defined as $t^B(G)=\min \left\{\frac{\mathrm{\mid }S\mathrm{\mid }}{c(G-S)}\right\}$, where the minimum is taken over all proper subsets $S\subset X$ (or $S\subset Y$) such that $c(G-S)>1$. In this paper, we give a sharp spectral radius condition for balanced bipartite graphs $G$ with $t^B(G)\ge 1$ to guarantee that $G$ contains Hamilton cycles. This solves a problem proposed in \cite{CFL}.