Fair division is the problem of allocating a set of items among agents in a fair manner. One of the most sought-after fairness notions is envy-freeness (EF), requiring that no agent envies another's allocation. When items are indivisible, it ceases to exist, and envy-freeness up to any good (EFX) emerged as one of its strongest relaxations. The existence of EFX allocations is arguably the biggest open question within fair division. Recently, Christodoulou, Fiat, Koutsoupias, and Sgouritsa (EC 2023) introduced showed that EFX allocations exist for the case of graphical valuations where an instance is represented by a graph: nodes are agents, edges are goods, and each agent values only her incident edges. On the other hand, they showed NP-hardness for checking the existence of EFX orientation where every edge is allocated to one of its incident vertices, and asked for a characterization of graphs that exhibit EFX orientation regardless of the assigned valuations. In this paper, we make significant progress toward answering their question. We introduce the notion of strongly EFX orientable graphs -- graphs that have EFX orientations regardless of how much agents value the edges. We show a surprising connection between this property and the chromatic number of the graph, namely $\chi(G)$ for graph $G$. In particular, we show that graphs with $\chi(G)\le 2$ are strongly EFX orientable, and those with $\chi(G)>3$ are not strongly EFX orientable. We provide examples of strongly EFX orientable and non-strongly EFX orientable graphs of $\chi(G)=3$ to prove tightness. Finally, we give a complete characterization of strong EFX orientability when restricted to binary valuations.

翻译：公平分配问题是将一组物品以公平方式分配给多个智能体的问题。最受关注的公平性概念之一是嫉妒无（EF）性质，要求没有智能体嫉妒他人的分配。当物品不可分割时，该性质不再成立，而“至多任意物品的嫉妒无”（EFX）成为其最强松弛之一。EFX分配的存在性无疑是公平分配领域最大的开放问题。近期，Christodoulou、Fiat、Koutsoupias与Sgouritsa（EC 2023）引入并证明了在图形估值情形下EFX分配的存在性：其实例由一张图表示，节点为智能体，边为物品，且每个智能体仅估值其邻接边。另一方面，他们证明了判定是否存在EFX定向（即每条边分配给其一个端节点）的NP困难性，并请求刻画无论估值如何均能展现EFX定向的图结构。本文在回答该问题方面取得了重要进展。我们引入强EFX可定向图的概念——无论智能体对边估值如何，均存在EFX定向的图。我们发现了该性质与图的色数（对于图$G$记作$\chi(G)$）之间的惊人联系。特别地，我们证明$\chi(G)\le 2$的图是强EFX可定向的，而$\chi(G)>3$的图则不是。我们给出$\chi(G)=3$的强EFX可定向与非强EFX可定向图实例以证明紧致性。最后，我们给出了二元估值情形下强EFX可定向性的完整刻画。