List-Avoiding Orientations

Given a graph G with a set F(v) of forbidden values at each \(v \in V(G)\), an F-avoiding orientation of G is an orientation in which \(\deg ^+(v) \not \in F(v)\) for each vertex v. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if \(|F(v)| < \frac{1}{2} \deg (v)\) for each \(v \in V(G)\), then G has an F-avoiding orientation, and they showed that this statement is true when \(\frac{1}{2}\) is replaced by \(\frac{1}{4}\). In this paper, we take a step toward this conjecture by proving that if \(|F(v)| < \lfloor \frac{1}{3} \deg (v) \rfloor \) for each vertex v, then G has an F-avoiding orientation. Furthermore, we show that if the maximum degree of G is subexponential in terms of the minimum degree, then this coefficient of \(\frac{1}{3}\) can be increased to \(\sqrt{2} - 1 - o(1) \approx 0.414\). Our main tool is a new sufficient condition for the existence of an F-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.