Hamilton cycles in (2, odd, 3)-Cayley graphs

Henry Glover (Avtor), Klavdija Kutnar (Avtor), Aleksander Malnič (Avtor), Dragan Marušič (Avtor)

Povzetek

In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups.

Ključne besede

Cayley graph;Hamilton cycle;arc-transitive graph;1-regular action;automorphism group;

Podatki

Jezik:	Angleški jezik
Leto izida:	2012
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	UP - Univerza na Primorskem
UDK:	519.17
COBISS:	1024390740
ISSN:	0024-6115
Št. ogledov:	2841
Št. prenosov:	133
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Sekundarni jezik:	Angleški jezik
Vrsta dela (COBISS):	Delo ni kategorizirano
Strani:	str. 1171-1197
Letnik:	ǂVol. ǂ104
Zvezek:	ǂno. ǂ6
Čas izdaje:	2012
ID:	1477173