A Family of One-regular Graphs of Valency 4

@article{Marusic1997AFO,
  title={A Family of One-regular Graphs of Valency 4},
  author={Dragan Marusic},
  journal={Eur. J. Comb.},
  year={1997},
  volume={18},
  pages={59-64}
}
A graph is said to beone-regularif its automorphism group acts regularly on the set of its arcs. A construction of an infinite family of one-regular graphs of valency 4 with vertex stabilizerZ22having a non-solvable group of automorphisms is given. The smallest graph in this family has 60 vertices. 
Constructing one-regular graphs of valency 4k with non-cyclic vertex stabilizer
A Family Of Tetravalent One-Regular Graphs
TLDR
In this paper, 4-valent one-regular graphs of order 5p, where p is a prime, are classified.
Constructing Infinite One-regular Graphs
TLDR
A construction of an infinite family of infinite one-regular graphs of valency 4 is given, which are Cayley graphs of almost abelian groups and hence of polynomial growth.
CUBIC SYMMETRIC GRAPHS OF ORDER 10p3
An automorphism group of a graph is said to be -regular if it acts regularly on the set of -arcs in the graph. A graph is -regular if its full automorphism group is -regular. In the present paper,
Tetravalent one-regular graphs of order 2pq
A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this article a complete classification of tetravalent one-regular graphs of order twice a product of two
CUBIC SYMMETRIC GRAPHS OF ORDER 10p
. An automorphism group of a graph is said to be s -regular if it acts regularly on the set of s -arcs in the graph. A graph is s -regular if its full automorphism group is s -regular. In the present
4-valent Graphs of Order 6p2 Admitting a Group of Automorphisms Acting Regularly on Arcs
TLDR
The 4 -valent graphs having 6 p 2 vertices, with p a prime, are classified, admitting a group of automorphisms acting regularly on arcs, and as a corollary, the 4-valent one-regular graphs are obtained.
A classification of tetravalent one-regular graphs of order $3p^2$
A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, tetravalent one-regular graphs of order 3p, where p is a prime,
...
...

References

SHOWING 1-10 OF 12 REFERENCES
Remarks on Path-transitivity in Finite Graphs
TLDR
It is shown that if the automorphism group of a graph Γ is transitive on vertices and on undirected paths of lengthk+1 in Γ, for somek≥1, then the group is alsotransitive onk-arcs inΓ.
A one-regular graph of degree three
Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to
On 4-Valent Symmetric Graphs
TLDR
This paper gives a complete classification of the graphs arising in (a) when the normal subgroup N is elementary abelian and (b), which depends to some extent on case (a), is more technical and is studied in a subsequent paper.
Constructing graphs which are ½-transitive
Abstract An infinite family of vertex-and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed.
Random Permutations: Some Group-Theoretic Aspects
The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group
...
...