A 2-connected graph is hamiltonian if every pair of nonadjacent vertices at distance two has a vertex of degree at least the bipartite independence number; a 3-connected analogue gives hamiltonian-connectedness.
Fan, New sufficient conditions for cycles in graphs, J
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A Fan-type condition involving bipartite independence number for hamiltonicity in graphs
A 2-connected graph is hamiltonian if every pair of nonadjacent vertices at distance two has a vertex of degree at least the bipartite independence number; a 3-connected analogue gives hamiltonian-connectedness.