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13.
Consistent cycles in graphs and digraphs
Miklavič, Štefko ; Potočnik, Primož, 1971- ; Wilson, Steve, matematik
Naj bo ▫$\Gamma$▫ končen digraf in naj bo ▫$G$▫ grupa avtomorfizmov grafa ▫$\Gamma$▫. Usmerjeni cikel ▫$\vec{C}$▫ digrafa ▫$\Gamma$▫ se imenuje ▫$G$▫-konsistenten, če obstaja takšen element grupe ...▫$G$▫, katerega zožitev na ▫$\vec{C}$▫ je rotacija ▫$\vec{C}$▫ (za en korak). Konsistentne cikle ločno-tranzitivnih grafov je prvi vpeljal J. H. Conway na svojem predavanju na "Second British Combinatorial Conference" leta 1971. Opazil je, da je v primeru, ko je grupa ▫$G$▫ ločno tranzitivna, število ▫$G$▫-orbit ▫$G$▫-konsistentnih ciklov natanko za ena manjše od stopnje grafa $\Gamma$. V članku dokažemo ta rezultat v splošnejši situaciji, ko je ▫$\Gamma$▫ digraf, grupa ▫$G$▫ pa poljubna.
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