Sum-list-colouring of theta-hypergraphs - Drgas-Burchardt, Ewa (Ars mathematica contemporanea, 2022)
Given a hypergraph ?$\mathcal{H}$? and a function ?$f:V( \mathcal{H}) → \mathbb{N}$?, we say that ?$\mathcal{H}$? is ?$f$?-choosable if there is a proper vertex coloring ?$φ$? of ?$\mathcal{H}$? such that ?$φ(v) \in L(v)$? for all ?$v\in V(\mathcal{H})$?, where ?$L:V( \mathcal{H}) → 2^{\mathbb{N}}$? is any assignment of ?$f(v)$? colors to a vertex ?$v$?. The sum choice number ?$χ_{sc}(\mathcal{H})$? of ?$\mathcal{H}$? is defined to be the minimum of ?$\sum_{v \in V(\mathcal{H})} f(v)$? over all functions ?$f$? such that ?$\mathcal{H}$? is ?$f$?-choosable. A trivial upper bound on ?$χ_{sc} (\mathcal{H})$? is ?$|V( \mathcal{H}|) + | \mathcal{E}(\mathcal{H})|$?. The class ?$Γ_{sc}$? of hypergraphs that achieve this bound is induced hereditary. We analyze some properties of hypergraphs in ?$Γ_{sc}$? as well as properties of hypergraphs in the class of forbidden hypergraphs for ?$Γ_{sc}$?. We characterize all ?$θ $?-hypergraphs in ?$Γ_{sc}$?, which leads to the characterization of all ?$θ$?-hypergraphs that are forbidden for ?$Γ_{sc}$?.
Keywords: ?$θ $?-hipergrafi; ?$θ $?-hypergraphs; hipergrafi; hypergraphs; sum-llist-colouring; vsotno seznamsko barvanje