2007
DOI: 10.1016/j.jctb.2006.07.008
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Regular t-balanced Cayley maps

Abstract: The concept of a t-balancedCayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Širáň, M. Škoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Širáň, Skewmorphisms of regular Cayley maps, Discrete Math. 244 (1-3) (2002) 167-179], and apply our results to the… Show more

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Cited by 39 publications
(47 citation statements)
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“…A regular Cayley map is called anti-balanced [22] if the power function of its skew morphism has precisely the two values 1 and −1. We will show in the next section that A = Z 2 × Z 4n has no balanced regular Cayley map for n > 1, but the following result from [3] shows A does have an anti-balanced one. Theorem 2.6 There are exactly three types of anti-balanced regular Cayley maps for finite abelian groups.…”
Section: And the Orbit Of C Generates A The Orbit Of X Generates D(a)mentioning
confidence: 95%
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“…A regular Cayley map is called anti-balanced [22] if the power function of its skew morphism has precisely the two values 1 and −1. We will show in the next section that A = Z 2 × Z 4n has no balanced regular Cayley map for n > 1, but the following result from [3] shows A does have an anti-balanced one. Theorem 2.6 There are exactly three types of anti-balanced regular Cayley maps for finite abelian groups.…”
Section: And the Orbit Of C Generates A The Orbit Of X Generates D(a)mentioning
confidence: 95%
“…For details, see [3]. It is an interesting exercise to check in each case that f is a skew morphism, with power function 1 on B and −1 on B + x.…”
Section: And the Orbit Of C Generates A The Orbit Of X Generates D(a)mentioning
confidence: 99%
“…Then, C and E are in the stabilizer of the point [1,0] t in PSL 2 (2 r ) and the order of C is equal to that of x, hence it is odd. Note that two elements of odd order in a dihedral group or a dicyclic group should commute.…”
Section: Case 5 : H(κ) Is Isomorphic To a Projective Group H With Pslmentioning
confidence: 99%
“…One can easily check CE = EC. Thus, the stabilizer of the point [1,0] t in PSL 2 (2 r ) is isomorphic to neither an abelian group, a dihedral group nor a dicyclic group. Therefore, in all cases, κ is either the identity or…”
Section: Case 5 : H(κ) Is Isomorphic To a Projective Group H With Pslmentioning
confidence: 99%
“…Note that the classification of symmetric graphs of order pq (p = q) was completed by [14][15][16][17] and the classification of symmetric graphs of order 4p is still elusive. For more results on regular maps, we refer the reader to [4,[18][19][20][21][22][23][24][25][26][27][28][29].…”
mentioning
confidence: 99%