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13.
Hamiltonicity of cubic Cayley graphs
Glover, Henry ; Marušič, Dragan
Izhajajoč iz problema, ki ga je leta 1969 postavil Lovász, je zadnja leta vse bolj prisotna domneva, da ima vsak povezan točkovno tranzitivni graf Hamiltonsko pot. V tem članku se pravilnost te ...domneve dokaže za kubične Cayleyeve grafe, ki pripadajo grupam z ▫$(2,s,3)$▫-prezentacijo, to je, grupam ▫$G = \langle a,b| a^2=1, b^s=1, (ab)^3=1, \dots \rangle$▫, ki so generirane z involucijo ▫$a$▫ and elementom ▫$b$▫ reda ▫$s \geq 3$▫ takim, da je produkt ▫$ab$▫ reda ▫$3$▫. Natančneje povedano: dokaže se, da ima Cayleyev graf ▫$X = Cay(G,\{a,b,b^{-1}\})$▫ hamiltonski cikel, ko je moč ▫$|G|$▫ (in tedaj tudi ▫$s$▫) kongruentna ▫$2$▫ modulo ▫$4$▫, in ima dolg cikel, ki spusti le dve točki grafa (in zato tudi Hamiltonsko pot), ko je moč ▫$|G|$▫ konruentna ▫$0$▫ modulo ▫$4$▫.
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