Tree Path Labeling of Hypergraphs – A Generalization of the Consecutive Ones Property

Abstract

Given a set system $\mathcal{F} \subseteq$ (2^U ∖ ∅ ) of a finite set U of cardinality n and a tree T of size n, does there exist at least one bijection φ:U → V(T) such that for each $S \in \mathcal{F}$, the set {φ(x) |x ∈ S} is the vertex set of a path in T? Our main result is that the existence of such a bijection from U to V(T) is equivalent to the existence of a function $\mathcal{l}$ from $\mathcal{F}$ to the set of all paths in T such that for any three, not necessarily distinct, $S_1, S_2, S_3 \in \mathcal{F}$, $|S_1 \cap S_2 \cap S_3|=|\mathcal{l}(S_1) \cap \mathcal{l}(S_2) \cap \mathcal{l}(S_3)|$. $\mathcal{l}$ is referred to as a tree path labeling of $\mathcal{F}$.