389 KB45 KB200820252022Drgas-Burchardt, EwaDrzystek, AgataSidorowicz, Elżbietaštevilka:1letnik:22P1.05 (15 str.)DOI:10.26493/1855-3974.2083.e80COBISSID_HOST:115433219ISSN:1855-3966URN:URN:NBN:SI:doc-LMOY5732enUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporanea?$\theta $?-hipergrafi?$\theta $?-hypergraphshipergrafihypergraphssum-llist-colouringvsotno seznamsko barvanjeSum-list-colouring of theta-hypergraphs|Given a hypergraph ?$\mathcal{H}$? and a function ?$f:V( \mathcal{H}) \rightarrow \mathbb{N}$?, we say that ?$\mathcal{H}$? is ?$f$?-choosable if there is a proper vertex coloring ?$\phi$? of ?$\mathcal{H}$? such that ?$\phi(v) \in L(v)$? for all ?$v\in V(\mathcal{H})$?, where ?$L:V( \mathcal{H}) \rightarrow 2^{\mathbb{N}}$? is any assignment of ?$f(v)$? colors to a vertex ?$v$?. The sum choice number ?$\chi_{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$?-choosable. A trivial upper bound on ?$\chi_{sc} (\mathcal{H})$? is ?$|V( \mathcal{H}|) + | \mathcal{E}(\mathcal{H})|$?. The class ?$\Gamma_{sc}$? of hypergraphs that achieve this bound is induced hereditary. We analyze some properties of hypergraphs in ?$\Gamma_{sc}$? as well as properties of hypergraphs in the class of forbidden hypergraphs for ?$\Gamma_{sc}$?. We characterize all ?$\theta $?-hypergraphs in ?$\Gamma_{sc}$?, which leads to the characterization of all ?$\theta$?-hypergraphs that are forbidden for ?$\Gamma_{sc}$?Če sta dana hipergraf ?$\mathcal{H}$? in funkcija ?$f:V( \mathcal{H}) \rightarrow \mathbb{N} $?, pravimo, da je ?$\mathcal{H}$? ?$f$?-izbiren, če obstaja pravilno vozliščno barvanje ?$\phi$?, pri katerem je ?$\phi(v) \in L(v)$? za vsa vozlišča ?$v\in V(\mathcal{H})$?, kjer je ?$L:V( \mathcal{H}) \rightarrow 2^{\mathbb{N}}$? poljubna dodelitev ?$f(v)$? barv vozlišča ?$v$?. Vsotno izbirni indeks ?$\chi_{sc}(\mathcal{H})$? hipergrafa ?$\mathcal{H}$? je definiran kot minimum vsote ?$\sum_{v \in V(\mathcal{H})} f(v)$? po vseh funkcijah ?$f$?, za katere je ?$\mathcal{H}$? ?$f$?-izbiren. Trivialna zgornja meja za ?$\chi_{sc} (\mathcal{H})$? je ?$|V(\mathcal{H}|) + |\mathcal{E} \mathcal{H})|$?. Razred ?$\Gamma_{sc}$? hipergrafov, ki dosežejo to mejo, je induciran hereditarno. Analiziramo nekaj lastnosti hipergrafov iz razreda ?$\Gamma_{sc}$?, pa tudi lastnosti hipergrafov iz razreda prepovedanih hipergrafov za ?$\Gamma_{sc}$?. Karakteriziramo vse ?$\theta$?-hipergrafe iz razreda ?$\Gamma_{sc}$?, od tod pa dobimo tudi karakterizacijo vseh ?$\theta$?-hipergrafov, ki so prepovedani za ?$\Gamma_{sc}$?TEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije