Degree sum condition on distance 2 vertices for hamiltonian cycles in balanced bipartite graphs

Let G be a 2-connected balanced bipartite graph of order 2n with n > 3. Denote mu(G) = min{max{d(x), d(y)} : x, y e V (G), dist(x, y) = 2} and mu 2(G) = min{d(x) + d(y) : x, y e V (G), dist(x, y) = 2}. Wang and Liu (2018) [14] proved that if mu(G) > n+1 2 , then G is hamiltonian. In this paper, we characterize non-hamiltonian bipartite graphs with mu(G) = n2 . Also, we show that G is bipancyclic with one exception, if mu(G) > n+12 . As a direct consequence, we also show that if mu 2(G) > n + 1, then G is hamiltonian and, with one exception, G is bipancyclic. Moreover, if we replace n + 1 with n + 2, then G contains a hamiltonian cycle passing through every edge of a perfect matching. The lower bounds in all results are sharp.(c) 2023 Elsevier B.V. All rights reserved.