A local independence number condition for n-extendable graphs

Let G be a graph and v ∈ V(G). Let Nk(v) = {u | u ∈ V(G) and d(u, v) = k}. It is proved that if G is a connected graph with ∞ > g(G) ⩾ 4 and with even order and if, for each vertex v in V(G), α(G[N2(v)]) ⩽ d(v) − 1, then G is regular and ⌈d(v)/4⌉-extendable. All results in this paper are sharp.