New Local Conditions for a Graph to be Hamiltonian

Abstract.For a vertex w of a graph G the ball of radius 2 centered at w is the subgraph of G induced by the set M2(w) of all vertices whose distance from w does not exceed 2. We prove the following theorem: Let G be a connected graph where every ball of radius 2 is 2-connected and d(u)+d(v)≥|M2(w)|−1 for every induced path uwv. Then either G is hamiltonian or for some p≥2 where ∨ denotes join.As a corollary we obtain the following local analogue of a theorem of Nash-Williams: A connected r-regular graph G is hamiltonian if every ball of radius 2 is 2-connected and for each vertex w of G.