Sum-list-colouring of θ-hypergraphs

Given a hypergraph ℋ and a function f :  V (ℋ) → ℕ , we say that ℋ is f -choosable if there is a proper vertex coloring ϕ of ℋ such that ϕ ( v ) ∈  L ( v ) for all v  ∈  V (ℋ) , where L :  V (ℋ) → 2 ℕ is any assignment of f ( v ) colors to a vertex v . The sum choice number χ s c (ℋ) of ℋ is defined to be the minimum of ∑ v  ∈  V (ℋ) f ( v ) over all functions f such that ℋ is f -choosable. A trivial upper bound on χ s c (ℋ) is | V (ℋ)| + |ℰ(ℋ)| . The class Γ s c of hypergraphs that achieve this bound is induced hereditary. We analyze some properties of hypergraphs in Γ s c as well as properties of hypergraphs in the class of forbidden hypergraphs for Γ s c . We characterize all θ -hypergraphs in Γ s c , which leads to the characterization of all θ -hypergraphs that are forbidden for Γ s c .