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Tetravalent graphs admitting half-transitive group actions: alternating cycles
Marušič, Dragan ; Praeger, Cheryl E.
V tem članku študiramo končne, povezane 4-valentne grafe ▫$X$▫, ki dopuščajo delovanje grupe ▫$G$▫, ki je tranzitivna na točkah in povezavah, ni pa tranzitivna na usmerjenih povezavah. Takemu grafu ...▫$X$▫ pravimo ▫$(G,1/2)$▫-tranzitiven. Grupa ▫$G$▫ inducira usmeritev povezav grafa ▫$X$▫. Tako je mogoče določiti posebno množico tako imenovanih alternirajočih ciklov, to je ciklov z nasprotno usmeritvijo povezav s skupnim krajiščem. Dokažemo, da je ▫$X$▫, krov končnega, povezanega 4-valentnega ▫$(G,1/2)$▫-tranzitivnega grafa, v katerem imajo alternirajoči cikli eno od naslednjih treh dodatnih lastnosti in so: tesno speti, ohlapno speti oziroma nasprotno speti. Podamo primere grafov z vsako od teh posebnih lastnosti spetosti in dokončno klasifikacijo tesno spetih grafov.
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