1997
DOI: 10.1002/(sici)1097-0118(199706)25:2<133::aid-jgt5>3.0.co;2-n
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A �-transitive graph of valency 4 with a nonsolvable group of automorphisms

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Cited by 49 publications
(10 citation statements)
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“…By Cheng and Oxley's classification of symmetric graphs of order 2p [5], Miller's construction is actually the all cubic 1-regular graphs of order 2p. Marušič and Xu [42] showed a way to construct a cubic 1-regular graph Y from a tetravalent half-transitive graph X with girth 3 by letting the triangles of X be the vertices in Y with two triangles being adjacent when they share a common vertex in X. Using the Marušič and Xu's result, Miller's construction can be generalized to graphs of order 2n, where n 13 is odd such that 3 divides ϕ(n), the Euler function (see [1,41]).…”
Section: Introductionmentioning
confidence: 99%
“…By Cheng and Oxley's classification of symmetric graphs of order 2p [5], Miller's construction is actually the all cubic 1-regular graphs of order 2p. Marušič and Xu [42] showed a way to construct a cubic 1-regular graph Y from a tetravalent half-transitive graph X with girth 3 by letting the triangles of X be the vertices in Y with two triangles being adjacent when they share a common vertex in X. Using the Marušič and Xu's result, Miller's construction can be generalized to graphs of order 2n, where n 13 is odd such that 3 divides ϕ(n), the Euler function (see [1,41]).…”
Section: Introductionmentioning
confidence: 99%
“…The first example of cubic one-regular graph was constructed by Frucht [5] with 432 vertices and lots of work have been done on cubic one-regular graphs as part of a more general problem dealing with the investigation of cubic arc-transitive graphs (see [6][7][8]). In 1997, Marušič and Xu [9] showed a way to construct a cubic one-regular graph Y from a tetravalent half-arc-transitive graph X with girth 3 by letting the triangles of X be the vertices in Y with two triangles being adjacent when they share a common vertex in X. Thus, one can construct infinite many cubic one-regular graphs from the infinite family of tetravalent half-arc-transitive graphs with girth 3 constructed by Alspach et al in [10] and from another infinite family of tetravalent half-arc-transitive graphs with girth 3 constructed by Marušič and Nedela in [11].…”
Section: Introductionmentioning
confidence: 99%
“…The first 1-regular cubic graph was constructed by Frucht [5] and later Miller [6] constructed infinitely many 1-regular cubic graphs of girth 6. Marušič and Xu [7] showed a way to construct a 1-regular cubic graph Y from a tetravalent halftransitive graph X with girth 3 by letting the triangles of X be the vertices in Y with two triangles being adjacent when they share a common vertex in X, so that one may construct infinitely many 1-regular cubic graphs from the tetravalent half-transitive graphs of girth 3 given in refs. [8,9].…”
Section: Introductionmentioning
confidence: 99%