386 KB40 KB200820252022Tyc, Adamštevilka:1letnik:22P1.02 (str. 23-39)DOI:10.26493/1855-3974.2242.842COBISSID_HOST:115140867ISSN:1855-3966URN:URN:NBN:SI:doc-RWRF3E4IenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneacikcak črtadirected Eulerian embeddingtriangulacija ploskvetriangulation of a surfaceusmerjena evklidska vložitevzigzagz-monodromijaz-monodromyz-orientacijaz-orientationZ-oriented triangulations of surfaces|The main objects of the paper are ?$z$?-oriented triangulations of connected closed 2-dimensional surfaces. A ?$z$?-orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an edge -- zigzags from the ?$z$?-orientation pass through this edge in different directions (type I) or in the same direction (type II). Then there are two types of faces in a triangulation: the first type is when two edges of the face are of type I and one edge is of type II and the second type is when all edges of the face are of type II. We investigate ?$z$?-oriented triangulations with all faces of the first type (in the general case, any ?$z$?-oriented triangulation can be shredded to a ?$z$?-oriented triangulation of such type). A zigzag is homogeneous if it contains precisely two edges of type I after any edge of type II. We give a topological characterization of the homogeneity of zigzags; in particular, we describe a one-to-one correspondence between ?$z$?-oriented triangulations with homogeneous zigzags and closed 2-cell embeddings of directed Eulerian graphs in surfaces. At the end, we give an application to one type of the ?$z$?-monodromyGlavni predmet članka so ?$z$?-orientirane triangulacije povezanih 2-dimenzionalnih ploskev. ?$z$?-orientacija zemljevida je minimalna kolekcija cikcak črt, ki dvojno pokriva množico povezav. Imamo dva tipa povezav – cikcak črte iz ?$z$?-orientacije potekajo skozi dano povezavo v različnih smereh (tip I) ali pa v isti smeri (tip II). V skladu s tem obstajata dva tipa lic v triangulaciji: pri prvem tipu sta dve povezavi lica tipa I, ena povezava pa tipa II, pri drugem tipu pa so vse povezave lica tipa II. Preučujemo ?$z$?-orientirane triangulacije z vsemi lici prvega tipa (v splošnem primeru velja, da se da poljubna ?$z$?-orientirana triangulacija razrezati na ?$z$?-orientirano triangulacijo tega tipa). Cikcak črta je homogena, če vsaki povezavi tipa II sledita natanko dve povezavi tipa I. Podamo topološko karakterizacijo homogenosti cikcak črt; posebej, opišemo bijektivno korespondenco med z-orientiranimi triangulacijami s homogenimi cikcak črtami in zaprtimi 2-celičnimi vložitvami usmerjenih Eulerjevih grafov v ploskve. Na koncu predstavimo aplikacijo na en tip ?$z$?-monodromijeTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije