The long-standing Erdős–Faber–Lovász conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using n colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős–Faber–Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.

长期存在的Erdős–Faber–Lovász猜想指出，具有n条边的每个n均匀线性超图形都具有使用n种颜色的适当顶点着色。在本文中，我们为该问题提出了一个代数框架，并提出了一个相应的更强的猜想。使用组合Nullstellensatz，我们将Erdős-Faber-Lovász猜想简化为某些多项式中存在非零系数。这些系数又与某些辅助图的规定度数顺序的方向数量有关。我们证明了某些取向的存在，这证明了我们的代数方法工作的必要条件。