Strong orientation of a connected graph for a crossing family

Abstract

Given a connected graph $G=(V,E)$ and a crossing family $C$ over ground set V such that $\left|{\delta }_G\right(U\left)\right|\ge 2$ for every $U\in C$, we prove there exists a strong orientation of G for $C$, i.e., an orientation of G such that each set in $C$ has at least one outgoing and at least one incoming arc. This implies the main conjecture in Chudnovsky et al. (2016) [3].