Locally dense supereulerian digraphs

Let λ(D) be the arc strong-connectivity of a digraph D, and k>0 be an integer, and α,β be rational numbers. A strong digraph D is locally (α,β)+-arc-connected if ∀v∈V(D), λ(D[N+(v)])≥α|N+(v)|+β. A locally (0,k)+-arc-connected digraph is also called k+-locally-arc-connected. We show that for any integer k, a strong, k+-locally-arc-connected digraph may not be supereulerian, and we also show that every locally (23,0)+-arc-connected strong digraph is supereulerian.