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.

给定一个图 G，在每个 \(v \in V(G)\) 处都有一组禁止值 F(v)，G 的 F 避免方向是一个方向，其中 \(\deg ^+(v) \对于每个顶点 v，不是 \in F(v)\)。Akbari、Dalirrooyfard、Ehsani、Ozeki 和 Sherkati 推测，如果 \(|F(v)| < \frac{1}{2} \deg (v)\ ) 对于每个 \(v \in V(G)\)，则 G 具有 F 避免方向，并且他们表明，当 \(\frac{1}{2}\) 被替换为 \( \压裂{1}{4}\)。在本文中，我们向这个猜想迈出了一步，证明如果对于每个顶点 v，如果 \(|F(v)| < \lfloor \frac{1}{3} \deg (v) \rfloor \)，则 G具有 F 回避倾向。此外，我们表明，如果 G 的最大次数相对于最小次数而言是次指数，那么 \(\frac{1}{3}\) 的系数可以增加到 \(\sqrt{2} - 1 - o(1)\约0.414\)。我们的主要工具是基于 Alon 和 Tarsi 的组合 Nullstellensatz 的 F 回避方向存在的新充分条件。