Observations on Woodall’s Conjecture

Let D = (V,A) be a directed graph. For U ⊆ V , δin(U) and δout(U) denote the sets of arcs entering U and leaving U , respectively. Moreover, δ(U) := δin(U) ∪ δout(U), din(U) := |δin(U)| (the indegree), dout(U) := |δout(U)| (the outdegree), and d(U) := |δ(U)| (the degree or total degree). If U = {u} is a singleton, we replace the argument {u} by u. We attach subscript D or A if useful. For B ⊆ A, B−1 := {(u, v) | (v, u) ∈ B}. A directed cut is a subset C of A such that C = δin(U) for some subset U of V satisfying ∅ 6= U 6= V and δout(U) = ∅. We say that U determines a directed cut if U is a subset of V satisfying ∅ 6= U 6= V and δout(U) = ∅. Denote by σ(D) the minimum size of a directed cut. This is ∞ if D has no directed cut, i.e., if D is strongly connected. A directed cut cover or dijoin is a subset B of A intersecting each directed cut. Trivially, B is a directed cut cover if and only if the digraph (V,A∪B−1) is strongly connected. Call a subset B of A strengthening if the digraph (V, (A \ B) ∪ B−1) is strongly connected. So each strengthening arc set is a directed cut cover. Call a function φ : A → [k] a strong coloring or strong k-coloring if φ−1(i) is strengthening for each i ∈ [k].