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

Glover, Henry; Kutnar, Klavdija; Malnič, Aleksander; Marušič, Dragan

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Title:	Hamilton cycles in (2, odd, 3)-Cayley graphs
Authors:	ID Glover, Henry (Author) ID Kutnar, Klavdija (Author) ID Malnič, Aleksander (Author) ID Marušič, Dragan (Author)
Files:	http://dx.doi.org/10.1112/plms/pdr042
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	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.
Keywords:	Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group
Year of publishing:	2012
Number of pages:	str. 1171-1197
Numbering:	Vol. 104, no. 6
PID:	20.500.12556/RUP-3442
ISSN:	0024-6115
UDC:	519.17
COBISS.SI-ID:	1024390740
Publication date in RUP:	15.10.2013
Views:	3993
Downloads:	136
Metadata:
:	GLOVER, Henry, KUTNAR, Klavdija, MALNIČ, Aleksander and MARUŠIČ, Dragan, 2012, Hamilton cycles in (2, odd, 3)-Cayley graphs. [online]. 2012. Vol. 104, no. 6, p. 1171–1197. [Accessed 8 April 2025]. Retrieved from: http://dx.doi.org/10.1112/plms/pdr042 Copy citation