One sufficient condition for Hamiltonian graphs involving distances
Kewen Zhao, Yue Der Lin, Ping Zhang · Russian Mathematics · 2012
In 1990 G. T. Chen proved that if G is a 2-connected graph of order n and 2| N ( x ) ∪ N ( y )| + d ( x ) + d ( y ) ≥ 2 n − 1 for each pair of nonadjacent vertices x, y ∈ V ( G ), then G is Hamiltonian. In this paper we prove that if G is a 2-connected graph of order n and 2| N ( x ) ∪ N ( y )| + d ( x )+ d ( y ) ≥ 2 n −1 for each pair of nonadjacent vertices x, y ∈ V ( G ) such that d ( x, y ) = 2, then G is Hamiltonian.