Peeling Hypergraphs: A Structural Sufficient Condition for Property B

We introduce the class of \emph{peeling hypergraphs}, defined by the existence of a linear order of the edges along which each edge still possesses enough untouched vertices to be 2‑coloured greedily. Every peeling hypergraph has Property~B, and the condition is purely structural, in contrast with probabilistic or algebraic approaches. We prove that peelability admits an equivalent characterisation as the existence of an acyclic first‑occurrence assignment of vertices to edges, which reveals the combinatorial core of the notion. The class is closed under disjoint unions and pendant edge additions, and for ordinary graphs it coincides with forests, thus generalising graph acyclicity. All cycles are minimal non‑peeling hypergraphs. We provide several examples, discuss the relationship with other hypergraph properties such as $\beta$-acyclicity, and analyse a greedy algorithm that attempts to build a peelable ordering. Several open problems concerning recognition complexity, minimal obstructions, and the generalisation to strong $k$-colouring are proposed.