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.
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Asratian, A. New Local Conditions for a Graph to be Hamiltonian. Graphs and Combinatorics 22, 153–160 (2006). https://doi.org/10.1007/s00373-006-0646-3
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DOI: https://doi.org/10.1007/s00373-006-0646-3