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

Abstract

Let G be a 2-connected balanced bipartite graph of order 2n with n≥3. Denote μ(G)=min⁡{max⁡{d(x),d(y)}:x,y∈V(G),dist(x,y)=2} and μ2(G)=min⁡{d(x)+d(y):x,y∈V(G),dist(x,y)=2}. Wang and Liu (2018) [14] proved that if μ(G)≥n+12, then G is hamiltonian. In this paper, we characterize non-hamiltonian bipartite graphs with μ(G)=n2. Also, we show that G is bipancyclic with one exception, if μ(G)≥n+12. As a direct consequence, we also show that if μ2(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.