Cubic vertex‐transitive graphs of order 2pq

A graph is vertex?transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1?S and S={s^?1 | s∈S}. The Cayleygraph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g∈G, s∈S}. Feng and Kwak [J Combin Theory B 97 (2007), 627–646; J Austral Math Soc 81 (2006), 153–164] classified all cubic symmetric graphs of order 4p or 2p² and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex‐transitive non‐Cayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex‐transitive graphs of order 2pq can be deduced. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 285–302, 2010     

Constructing cubic edge‐ but not vertex‐transitive graphs

An infinite family of cubic edge‐transitive but not vertex‐transitive graphs with edge stabilizer isomorphic to ℤ₂ is constructed. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 152–160, 2000     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Large Cayley graphs and vertex‐transitive non‐Cayley graphs of given degree and diameter

For any d?5 and k?3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d?3)/3)^k. By comparison with other available results in this area we show that our family gives the largest currently known Cayley graphs for a wide range of sufficiently large degrees and diameters. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 87–98, 2010     

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.     

Classification of edge‐transitive rose window graphs

Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}$ and edge set $\{\{{{x}}_{{i}},{{x}}_{{{i+1}}}\} \mid {{i}}\in {\mathbb{Z}}_n \} \cup \{\{{{y}}_{{{i}}},{{y}}_{{{i+r}}}\}\mid {{i}} \in{\mathbb{Z}}_{{n}}\}\cup \{\{{{x}}_{{{i}}},{{y}}_{{{i}}}\} \mid {{i}}\in {\mathbb{Z}}_{{{n}}}\}\cup \{\{{{x}}_{{{i+a}}},{{y}}_{{{i}}}\} \mid{{i}} \in {\mathbb{Z}}_{{{n}}}\}$. In this article a complete classification of edge‐transitive rose window graphs is given, thus solving one of the three open problems about these graphs posed by Steve Wilson in 2001. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 216–231, 2010     

Finite 2‐Geodesic Transitive Graphs of Prime Valency

We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc transitive or the valency p satisfies , and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph with automorphism group and diameter 3.     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Locally 2‐arc transitive graphs,homogeneous factorizations,and partial linear spaces

We establish natural bijections between three different classes of combinatorial objects; namely certain families of locally 2‐arc transitive graphs, partial linear spaces, and homogeneous factorizations of arc‐transitive graphs. Moreover, the bijections intertwine the actions of the relevant automorphism groups. Thus constructions in any of these areas provide examples for the others. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 139–148, 2006     

One‐factorizations of complete graphs with vertex‐regular automorphism groups

We consider one‐factorizations of K_2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one‐factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non‐existence statements in case the number of fixed one‐factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002     

The asphericity of random 2‐dimensional complexes

We study random 2‐dimensional complexes in the Linial–Meshulam model and prove that for the probability parameter satisfying a random 2‐complex Y contains several pairwise disjoint tetrahedra such that the 2‐complex Z obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex Z satisfies the Whitehead conjecture, i.e. any subcomplex is aspherical. This implies that Y is homotopy equivalent to a wedge where Z is a 2‐dimensional aspherical simplicial complex. We also show that under the assumptions where c > 3 and , the complex Z is genuinely 2‐dimensional and in particular, it has sizable 2‐dimensional homology; it follows that in the indicated range of the probability parameter p the cohomological dimension of the fundamental group of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.     

Semi‐transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi‐transitive and semi‐transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi‐transitive spider‐graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi‐transitive graph (if it has more than one) is a bi‐transitive graph. We show how the alternets can be used to understand the structure of a semi‐transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi‐transitive graphs of degree 4 begun by Marus?ic? and Praeger. This classification shows that nearly all such graphs are spider‐graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004     

Independent sets of maximal size in tensor powers of vertex‐transitive graphs

Let G be a connected, nonbipartite vertex‐transitive graph. We prove that if the only independent sets of maximal cardinality in the tensor product G × G are the preimages of the independent sets of maximal cardinality in G under projections, then the same holds for all finite tensor powers of G, thus providing an affirmative answer to a question raised by Larose and Tardif (J Graph Theory 40(3) (2002), 162–171). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 295‐301, 2009     

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Regular 4‐critical graphs of even degree

In 1960, Dirac posed the conjecture that r‐connected 4‐critical graphs exist for every r ≥ 3. In 1989, Erd?s conjectured that for every r ≥ 3 there exist r‐regular 4‐critical graphs. In this paper, a technique of constructing r‐regular r‐connected vertex‐transitive 4‐critical graphs for even r ≥ 4 is presented. Such graphs are found for r = 6, 8, 10. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 103–130, 2004     

Vertex‐Transitive Cubic Graphs of Square‐Free Order

A classification of connected vertex‐transitive cubic graphs of square‐free order is provided. It is shown that such graphs are well‐characterized metacirculants (including dihedrants, generalized Petersen graphs, Möbius bands), or Tutte's 8‐cage, or graphs arisen from simple groups PSL(2, p).