Strong orientation of a connected graph for a crossing family

TL;DR

This paper proves that for any connected graph with a crossing family satisfying certain edge conditions, there exists a strong orientation ensuring each crossing set has both incoming and outgoing edges, confirming a key conjecture.

Contribution

It establishes the existence of a strong orientation for graphs with crossing families, resolving the main conjecture in a prior study.

Findings

Existence of strong orientation for graphs with crossing families.

Implication that minimal counterexamples have disconnected arcs.

Supports the Edmonds-Giles conjecture in specific cases.

Abstract

Given a connected graph $G=(V,E)$ and a crossing family $\mathcal{C}$ over ground set $V$ such that $|\delta_G(U)|\geq 2$ for every $U\in \mathcal{C}$, we prove there exists a strong orientation of $G$ for $\mathcal{C}$, i.e., an orientation of $G$ such that each set in $\mathcal{C}$ has at least one outgoing and at least one incoming arc. This implies the main conjecture in Chudnovsky et al. (Disjoint dijoins. Journal of Combinatorial Theory, Series B, 120:18--35, 2016). In particular, in every minimal counterexample to the Edmonds-Giles conjecture where the minimum weight of a dicut is $2$, the arcs of nonzero weight must be disconnected.