On 1-Arc-regular Graphs

Fang, Xuekun; Wang, Jie; Xu, Ming

doi:10.1006/eujc.2002.0579

Cited by 25 publications

(15 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Problem A was partially considered in [5] for a connected cubic G-basic graph Γ , and in this paper we completely solve it (Theorem 1.1) for solvable groups, but it is still elusive when G is nonsolvable.…”

Section: Problem B Which Cubic Symmetric Graphs Are Cayley?mentioning

confidence: 99%

Symmetric cubic graphs with solvable automorphism groups

Feng

Zhou

2015

European Journal of Combinatorics

View full text Add to dashboard Cite

a b s t r a c tA cubic graph Γ is G-arc-transitive if G ≤ Aut(Γ ) acts transitively on the set of arcs of Γ , and G-basic if Γ is G-arc-transitive and G has no non-trivial normal subgroup with more than two orbits. Let G be a solvable group. In this paper, we first classify all connected G-basic cubic graphs and determine the group structure for every G. Then, combining covering techniques, we prove that a connected cubic G-arc-transitive graph is either a Cayley graph, or its full automorphism group is of type 2 2 , that is, a 2-regular group with no involution reversing an edge, and has a non-abelian normal subgroup such that the corresponding quotient graph is the complete bipartite graph of order 6.

show abstract

Section: Problem B Which Cubic Symmetric Graphs Are Cayley?mentioning

confidence: 99%

Symmetric cubic graphs with solvable automorphism groups

Feng

Zhou

2015

European Journal of Combinatorics

View full text Add to dashboard Cite

show abstract

“…Take a Sylow 5-subgroup, say P , of G, and set H = N G (P ). From Atlas [7, pp.4] it is easily known that H ∼ = D 10 . Note that all involutions in G are conjugate each other.…”

Section: Constructions Of Pentavalent Symmetric Graphsmentioning

confidence: 99%

“…Recently, Li [22] classified vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4, and Fang et al [8] classified vertex-primitive 2-regular graphs. For more results on symmetric graphs with general valencies, see, for example, [10,20,21,22]. Despite all of these efforts, however, further classifications of symmetric graphs with general valencies seem to be very difficult.…”

Section: Introductionmentioning

confidence: 99%

Pentavalent symmetric graphs of order 16p

Guo

Hou

Shi

2017

Acta Math. Appl. Sin. Engl. Ser.

View full text Add to dashboard Cite

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, a complete classification of connected pentavalent symmetric graphs of order 12p is given for each prime p. As a result, a connected pentavalent symmetric graph of order 12p exists if and only if p = 2, 3, 5 or 11, and up to isomorphism, there are only nine such graphs: one for each p = 2, 3 and 5, and six for p = 11.

show abstract

“…Denote by E = {B o , B\, ... , 6/_i) the set of orbits of N on V(X). Since N <l G, E is a complete block system of G. Consider the quotient graph X of X defined by V(X) = E and (B n fi 7 ) e E(X) if and only if there exist u, e B, and Vj e Bj such that (u,, Vj) e E(X). If N has more than two orbits, Lorimer [14,Theorem 9] showed that X is a cubic graph and G/N is a solvable one-regular subgroup of Aut(X) (also see [21]).…”

Section: Preliminariesmentioning

confidence: 99%

One-regular cubic graphs of order a small number times a prime or a prime square

Yan

Kwak

2004

J. Aust. Math. Soc.

View full text Add to dashboard Cite

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2p or 2p 2 where p is a prime if and only if 3 is a divisor of p -1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4p or 4p 2 .2000 Mathematics subject classification: primary O5C25, 20B25.

show abstract

On 1-Arc-regular Graphs

Cited by 25 publications

References 17 publications

Symmetric cubic graphs with solvable automorphism groups

Symmetric cubic graphs with solvable automorphism groups

Pentavalent symmetric graphs of order 16p

One-regular cubic graphs of order a small number times a prime or a prime square

Contact Info

Product

Resources

About