A spectral condition for Hamilton cycles in tough bipartite graphs - Ai, Lianyang; Zhang, Wenqian (Graphs and combinatorics, 2025)
Let G be a graph. The 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 bipartite toughness of G is defined as t B ( G ) = min | S | c ( G - S ) , where the minimum is taken over all proper subsets S ⊂ X (or S ⊂ 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 ) ≥ 1 to guarantee that G contains Hamilton cycles. This solves a problem proposed in [ 9 ].