A complete classification of cubic symmetric graphs of girth 6

Kutnar, Klavdija; Marušič, Dragan

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Title:	A complete classification of cubic symmetric graphs of girth 6
Authors:	ID Kutnar, Klavdija (Author) ID Marušič, Dragan (Author)
Files:	http://dx.doi.org/10.1016/j.jctb.2008.06.001
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	UPR - University of Primorska
Abstract:	A complete classification of cubic symmetric graphs of girth 6 is given. It is shown that with the exception of the Heawood graph, the Moebius-Kantor graph, the Pappus graph, and the Desargues graph, a cubic symmetric graph $X$ of girth 6 is a normal Cayley graph of a generalized dihedral group; in particular, (i) $X$ is 2-regular if and only if it is isomorphic to a so-called $I_k^n$ -path, a graph of order either $n^2/2$ or $n^2/6$ , which is characterized by the fact that its quotient relative to a certain semiregular automorphism is a path. (ii) $X$ is 1-regular if and only if there exists an integer $r$ with prime decomposition $r=3^s p_1^{e_1} \dots p_t^{e_t} > 3$ , where $s \in \{0,1\}$ , $t \ge 1$ , and $p_i \equiv 1 \pmod{3}$ , such that $X$ is isomorphic either to a Cayley graph of a dihedral group $D_{2r}$ of order $2r$ or $X$ is isomorphic to a certain $\ZZ_r$ -cover of one of the following graphs: the cube $Q_3$ , the Pappus graph or an $I_k^n(t)$ -path of order $n^2/2$ .
Keywords:	graph theory, cubic graphs, symmetric graphs, $s$ -regular graphs, girth, consistent cycle
Year of publishing:	2009
Number of pages:	str. 162-184
Numbering:	Vol. 99, No. 1
PID:	20.500.12556/RUP-1125
ISSN:	0095-8956
UDC:	519.17
COBISS.SI-ID:	2724823
Publication date in RUP:	15.10.2013
Views:	5006
Downloads:	89
Metadata:
:	KUTNAR, Klavdija and MARUŠIČ, Dragan, 2009, A complete classification of cubic symmetric graphs of girth 6. [online]. 2009. Vol. 99, no. 1, p. 162–184. [Accessed 12 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.jctb.2008.06.001 Copy citation