On the complexity of finding well-balanced orientations with upper bounds on the out-degrees
Résumé
Abstract We show that the problem of deciding whether a given graph G has a well-balanced orientation $$\vec {G}$$ G → such that $$d_{\vec {G}}^+(v)\le \ell (v)$$ d G → + ( v ) ≤ ℓ ( v ) for all $$v \in V(G)$$ v ∈ V ( G ) for a given function $$\ell :V(G)\rightarrow \mathbb {Z}_{\ge 0}$$ ℓ : V ( G ) → Z ≥ 0 is NP-complete. We also prove a similar result for best-balanced orientations. This improves a result of Bernáth, Iwata, Király, Király and Szigeti and answers a question of Frank.