5360 KB29 KB200820252019Jin, Xian'anYan, Qiletnik:17številka:2str. 637-652ISSN:1855-3966COBISSID_HOST:18976601URN:URN:NBN:SI:doc-XJJIT4MHenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaekstremalni minorextremal minormedial graphmedialni graforientacijaorientationPetrie dualPetriejev dualribbon graphtrakovni grafExtremal embedded graphs|Let ?$G$? be a ribbon graph and ?$\mu (G)$? be the number of components of the virtual link formed from ?$G$? as a cellularly embedded graph via the medial construction. In this paper we first prove that ?$\mu (G) \leq f(G) + \gamma (G)$?, where ?$f(G)$? and ?$\gamma (G)$? are the number of boundary components and Euler genus of ?$G$?, respectively. A ribbon graph is said to be extremal if ?$\mu (G) = f(G) + \gamma (G)$?. We then obtain that a ribbon graph is extremal if and only if its Petrial is plane. We introduce a notion of extremal minor and provide an excluded extremal minor characterization for extremal ribbon graphs. We also point out that a related result in the monograph by J. A. Ellis-Monaghan and I. Moffatt Graphs on surfaces. Dualities, polynomials, and knots. New York, NY: Springer (2013) is not correct and prove that two related conjectures raised by S. Huggett and I. Tawfik Ars Math. Contemp. 8, No. 2, 319-335 (2015) hold for more general ribbon graphsNaj bo ?$G$? trakovni graf in naj bo ?$\mu (G)$? število komponent virtualne povezave, tvorjene iz grafa ?$G$? kot celično vloženega grafa preko medialne konstrukcije. V članku najprej dokažemo, da je ?$\mu (G) \leq f(G) + \gamma (G)$?, kjer je ?$f(G)$? število robnih komponent, ?$\gamma (G)$? pa Eulerjev rod grafa ?$G$?. Trakovni graf je ekstremalen, če je ?$\mu (G) = f(G) + \gamma (G)$?. Nato pokažemo, da je trakovni graf ekstremalen, če in samo če je njegov Petrial ravninski graf. Vpeljemo pojem ekstremalnega minorja in podamo karakterizacijo ekstremalnih trakovnih grafov z izključenimi ekstremalnimi minorji. Izpostavimo tudi, da soroden rezultat v monografiji Ellis-Monaghana in Moffatta ni pravilen, ter dokažemo, da dve sorodni domnevi, ki sta ju postavila Huggett in Tawfik, veljata za splošnejše trakovne grafeTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije