Regular graph coverings whose covering transformation groups have the isomorphism extension property

Hong, Sungpyo; Kwak, Jin Ho; Lee, Jaeun

doi:10.1016/0012-365x(94)00266-l

Cited by 35 publications

(23 citation statements)

References 13 publications

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“…In the former case r − 3l = 4l and in the latter case r − 3l + b = 4l, both contradicting (12). Finally, if v = y b , then x r −3l and y b are adjacent by a 0-spoke, and so r − 3l = b, contradicting (12). Therefore x r −3l / ∈ V (C 1 ).…”

Section: Now Again Consider the 3-arcmentioning

confidence: 73%

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A classification of pentavalent arc-transitive bicirculants

2014

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A bicirculant is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. A graph is said to be arc-transitive if its automorphism group acts transitively on the set of its arcs. All cubic and tetravalent arc-transitive bicirculants are known, and this paper gives a complete classification of connected pentavalent arc-transitive bicirculants. In particular, it is shown that, with the exception of seven particular graphs, a connected pentavalent bicirculant is arc-transitive if and only if it is isomorphic to a Cayley graph Cay(D 2n , {b, ba, ba r +1 , ba r 2 +r +1 , ba r 3 +r 2 +r +1 }) on the dihedral group D 2n = a, b | a n = b 2 = baba = 1 , where r ∈ Z * n such that r 4 + r 3 + r 2 + r + 1 ≡ 0 (mod n).

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Section: Now Again Consider the 3-arcmentioning

confidence: 73%

“…The first one may be deduced from [20,Theorem 4.2], the second one from [12] whereas the third one is taken from [3, Proposition 2.2], but it may also be deduced from [21, Corollaries 9.4, 9.7, 9.8].…”

Section: Graph Coversmentioning

confidence: 99%

A classification of pentavalent arc-transitive bicirculants

2014

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“…It is easy to see that CK(2 m , 1, k) and CK(2 m , 2, k) are bipartite graphs. In fact, both are cyclic cover- The smallest graphs in the two infinite families of half-arc-transitive graphs given in Theorem 2.1 are CK (16,1) and CK (16,2), of which the first is depicted in Fig. 2.…”

Section: Main Theoremmentioning

confidence: 99%

“…Remark The smallest ones in the two infinite families of half-arc-transitive graphs CK(2 m , i) for i = 1, 2 given in Lemma 5.2 are CK (16,1) and CK (16,2), which have 128 vertices. We depict CK (16,1) z 4 = δ 1 + δ 2 + 1, z 5 = δ 2 , z 6 = δ 1 + 1, z 7 = δ 1 + δ 2 and z 9 = δ 1 , and if = −1 then z 3 = δ 1 , z 4 = δ 1 + δ 2 , z 5 = δ 2 , z 6 = δ 1 , z 7 = δ 1 + δ 2 + 1 and z 9 = δ 1 + 1.…”

Section: Lemma 51 If G 1 Lifts and X Ismentioning

confidence: 99%

“…We depict CK (16,1) z 4 = δ 1 + δ 2 + 1, z 5 = δ 2 , z 6 = δ 1 + 1, z 7 = δ 1 + δ 2 and z 9 = δ 1 , and if = −1 then z 3 = δ 1 , z 4 = δ 1 + δ 2 , z 5 = δ 2 , z 6 = δ 1 , z 7 = δ 1 + δ 2 + 1 and z 9 = δ 1 + 1. By Table 2, γ 3 can be extended to the automorphisms of Z 2 m induced by 1 → −1 for = 1 and by 1 → 1 for = −1, which correspond to columns 7 and 8 of Table 3.…”

Section: Lemma 51 If G 1 Lifts and X Ismentioning

confidence: 99%

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Constructing even radius tightly attached half-arc-transitive graphs of valency four

2007

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A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is called the radius of X. The graph X is said to be tightly attached if any two adjacent alternating cycles intersect in the same number of vertices equal to the radius of X. Marušič (J. Comb. Theory B, 73, 41-76, 1998) classified odd radius tightly attached tetravalent half-arctransitive graphs. In this paper, we classify the half-arc-transitive regular coverings of the complete bipartite graph K 4,4 whose covering transformation group is cyclic of prime-power order and whose fibre-preserving group contains a half-arc-transitive subgroup. As a result, two new infinite families of even radius tightly attached tetravalent half-arc-transitive graphs are constructed, introducing the first infinite families of tetravalent half-arc-transitive graphs of 2-power orders.

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s‐Regular cubic graphs as coverings of the complete bipartite graph K_3,3

Feng

Kwak

2003

Journal of Graph Theory

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A graph is s-regular if its automorphism group acts freely and transitively on the set of s-arcs. An infinite family of cubic 1-regular graphs was constructed in [10], as cyclic coverings of the three-dimensional Hypercube. In this paper, we classify the s-regular cyclic coverings of the complete bipartite graph K 3,3 for each s ! 1 whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed. ß

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Regular graph coverings whose covering transformation groups have the isomorphism extension property

Cited by 35 publications

References 13 publications

A classification of pentavalent arc-transitive bicirculants

A classification of pentavalent arc-transitive bicirculants

Constructing even radius tightly attached half-arc-transitive graphs of valency four

s‐Regular cubic graphs as coverings of the complete bipartite graph K_3,3

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Regular graph coverings whose covering transformation groups have the isomorphism extension property

Cited by 35 publications

References 13 publications

A classification of pentavalent arc-transitive bicirculants

A classification of pentavalent arc-transitive bicirculants

Constructing even radius tightly attached half-arc-transitive graphs of valency four

s‐Regular cubic graphs as coverings of the complete bipartite graph K3,3

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s‐Regular cubic graphs as coverings of the complete bipartite graph K_3,3