Vertex-transitive non-Cayley Graphs Research Articles

On cubic vertex-transitive non-Cayley graphs of 2-power orders

A Cayley graph (or bi-Cayley graph) Γ is a graph which admits a semiregular group G of automorphisms with exactly one orbit (or two orbits, respectively). Cayley graphs form an important class of vertex-transitive graphs. However, not all vertex-transitive graphs are Cayley. A typical example is the Petersen graph. In this paper, we first prove that all cubic vertex-transitive graphs of order 2-powers are bi-Cayley graphs. We then give a sufficient condition for a cubic vertex-transitive graph of order a 2-power to be non-Cayley. As an application, we construct an infinite family of cubic vertex-transitive non-Cayley graphs of order a 2-power.

A classification of connected cubic vertex-transitive bi-Cayley graphs over semidihedral group

A graph is said to be a bi-Cayley graph over a group H if there exists a subgroup of Aut(G) isomorphic to H acting semiregularly on its vertex set with two orbits. In this paper, we give a complete classification of connected cubic vertex-transitive bi-Cayley graphs over semidihedral group. As a byproduct, we construct a family of vertex-transitive non-Cayley graphs.

Cubic vertex-transitive non-Cayley graphs of order 12p

A graph is said to be vertex-transitive non-Cayley if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order 12p, where p is a prime, is given. As a result, there are 11 sporadic and one infinite family of such graphs, of which the sporadic ones occur when p equals 5, 7 or 17, and the infinite family exists if and only if p ≡ 1 (mod 4), and in this family there is a unique graph for a given order.

All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph DP(n,t), n≥3 and t∈Zn∖{0}, 2≤2t<n, has vertex-set {xi,yi,ui,vi∣i∈Zn}, edge-set {{xi,xi+1},{yi,yi+1},{ui,vi+t},{vi,ui+t},{xi,ui},{yi,vi}∣i∈Zn}. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of DP(n,t), and conjectured that all DP(n,t) are Hamiltonian. In this paper, we prove this conjecture.

Tetravalent Vertex-Transitive Graphs of Order Twice A Prime Square

A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, a complete classification is given of tetravalent vertex-transitive non-Cayley graphs of order $$2p^2$$2p2 for any prime p.

Super cyclically edge-connected vertex-transitive graphs of girth at least 5

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λc, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In [Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549–562 (2011)], it is proved that a connected vertex-transitive graph is super-λc if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λc graphs which have degree 4 and girth 5. In this paper, a characterization of k (k ≥ 4)-regular vertex-transitive nonsuper-λc graphs of girth 5 is given. Using this, we classify all k (k ≥ 4)-regular nonsuper-λc Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λc vertex-transitive non-Cayley graphs.

Cubic Vertex-Transitive Non-Cayley Graphs of Order $8p$

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertex-transitive non-Cayley graphs of order $8p$ are classified for each prime $p$. It follows from this classification that there are two sporadic and two infinite families of such graphs, of which the sporadic ones have order $56$, one infinite family exists for every prime $p&gt;3$ and the other family exists if and only if $p\equiv 1\mod 4$. For each family there is a unique graph for a given order.

Tetravalent vertex-transitive graphs of order 4p

A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this article, the tetravalent vertex-transitive non-Cayley graphs of order 4p are classified for each prime p. As a result, there are one sporadic and five infinite families of such graphs, of which the sporadic one has order 20, and one infinite family exists for every prime p>3, two families exist if and only if p≡1 (mod 8) and the other two families exist if and only if p≡1 (mod 4). For each family there is a unique graph for a given order. © 2011 Wiley Periodicals, Inc.

On non-Cayley vertex-transitive graphs and the meta-Cayley graphs

The pursuit to identify vertex-transitive non-Cayley graphs has been deliberate for some time now. In that vein, Alspach and Parsons [1] introduced metacirculant graphs. They are defined on two cyclic groups with adjacency resembling twisting that is typically used in defining semi-direct products of groups. In this sequel we generalise the construction to general groups and introduce a class of graphs we call meta-Cayley graphs.

A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions

We show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also prove that the graphs arising this way are hamiltonian provided that their valency is at least six.

Automorphisms of cubic Cayley graphs of order [formula omitted

In this paper the automorphism groups of connected cubic Cayley graphs of order 2 p q for distinct odd primes p and q are determined. As an application, all connected cubic non-symmetric Cayley graphs of order 2 p q are classified and this, together with classifications of connected cubic symmetric graphs and vertex-transitive non-Cayley graphs of order 2 p q given by the last two authors, completes a classification of connected cubic vertex-transitive graphs of order 2 p q .

On the number of closed walks in vertex-transitive graphs

The results of Širáň and the first author [A construction of vertex-transitive non-Cayley graphs, Australas. J. Combin. 10 (1994) 105–114; More constructions of vertex-transitive non-Cayley graphs based on counting closed walks, Australas. J. Combin. 14 (1996) 121–132] are generalized, and new formulas for the number of closed walks of length p r or pq, where p and q are primes, valid for all vertex-transitive graphs are found. Based on these formulas, several simple tests for vertex-transitivity are presented, as well as lower bounds on the orders of the smallest vertex- and arc-transitive groups of automorphisms for vertex-transitive graphs of given valence.

Finite edge-transitive Cayley graphs and rotary Cayley maps

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

Square-Free Non-Cayley Numbers. On Vertex-Transitive Non-Cayley Graphs of Square-Free Order

A complete classification is given of finite primitive permutation groups which contain a regular subgroup of square-free order. Then a collection $${\cal P}{\cal N}{\cal C}$$ of square-free numbers n is obtained such that there exists a vertex-primitive non-Cayley graph on n vertices if and only if n is a member of $${\cal P}{\cal N}{\cal C}$$ .

Triangle group representations and their applications to graphs and maps

The aim of this survey article is to draw the attention of the combinatorial community to representations of groups and their applications in the theory of graphs and maps. Specifically, we shall be interested in representations of the triangle groups 〈 y, z| y m = z n =( yz) 2=1〉 in special linear groups. Applications will include constructions of highly symmetrical finite maps of arbitrarily large planar width, Hurwitz groups, vertex-transitive non-Cayley graphs, and arc-transitive graphs of given valence and given exact girth.

A Note on Large Graphs of Diameter Two and Given Maximum Degree

Letvt(d, 2) be the largest order of a vertex-transitive graph of degreedand diameter 2. It is known thatvt(d, 2)=d2+1 ford=1, 2, 3, and 7; for the remaining values ofdwe havevt(d, 2)⩽d2−1. The only knowngenerallower bound onvt(d, 2), valid forall d, seems to bevt(d, 2)⩾⌊(d+2)/2⌋ ⌈(d+2)/2⌉. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows thatvt(d, 2)⩾(8/9)(d+12)2for alldof the formd=(3q−1)/2, whereqis a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, ford=7 we obtain as a special case the Hoffman–Singleton graph, and ford=11 andd=13 we have new largest graphs of diameter 2, and degreedon 98 and 162 vertices, respectively.

Vertex-Transitive Non-Cayley Graphs with Arbitrarily Large Vertex-Stabilizer

A construction is given for an infinite family {Γn} of finite vertex-transitive non-Cayley graphs of fixed valency with the property that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of Γn is a strictly increasing function ofn . For each n the graph is 4-valent and arc-transitive, with automorphism group a symmetric group of large prime degree \(p < 2^{2^n + 2}\). The construction uses Sierpinski's gasket to produce generating permutations for the vertex-stabilizer (a large 2-group).

Vertex-Primitive Graphs of Order a Product of Two Distinct Primes

Let k and p be odd primes with k < p. All vertex-primitive graphs of order kp are classified, and those which are symmetric, or are edge-transitive but not symmetric. or are not Cayley graphs are identified. In addition, a classification is given of all vertex-primitive, edge-transitive lantisymmetric) directed graphs or order kp. The classification of the vertex-primitive symmetric graphs of order kp is used by the authors and others to complete the classification of symmetric graphs or order kp. The classification of the vertex-primitive non-Cayley graphs of order kp is used by B.D. McKay and the first author in an investigation of vertex-transitive non-Cayley graphs.