On finite edge transitive graphs and rotary maps

It is shown that every connected vertex and edge transitive graph has a normal multicover that is a connected normal edge transitive Cayley graph. Moreover, every chiral or regular map has a normal cover that is a balanced chiral or regular Cayley map, respectively. As an application, a new family of half-transitive graphs is constructed as 2-fold covers of a family of 2-arc transitive graphs admitting Suzuki groups.     

An infinite family of cubic edge- but not vertex-transitive graphs

An infinite family of cubic edge- but not vertex-transitive graphs is constructed. The graphs are obtained as regular -covers of K_3,3 where n=p₁^e₁p₂^e₂p_k^e_k where p_i are distinct primes congruent to 1 modulo 3, and e_i1. Moreover, it is proved that the Gray graph (of order 54) is the smallest cubic edge- but not vertex-transitive graph.     

Homogeneous factorisations of complete graphs with edge-transitive factors

A factorisation of a complete graph K _n is a partition of its edges with each part corresponding to a spanning subgraph (not necessarily connected), called a factor. A factorisation is called homogeneous if there are subgroups M<G≤S _n such that M is vertex-transitive and fixes each factor setwise, and G permutes the factors transitively. We classify the homogeneous factorisations of K _n for which there are such subgroups G,M with M transitive on the edges of a factor as well as the vertices. We give infinitely many new examples. This paper forms part of an Australian Research Council Discovery Grant project, and was a major part of the PhD project of the second author.     

Characterization of digraphs with equal domination graphs and underlying graphs

A domination graph of a digraph D, dom(D), is created using the vertex set of D and edge {u,v}∈E[dom(D)] whenever (u,z)∈A(D) or (v,z)∈A(D) for every other vertex z∈V(D). The underlying graph of a digraph D, UG(D), is the graph for which D is a biorientation. We completely characterize digraphs whose underlying graphs are identical to their domination graphs, UG(D)=dom(D). The maximum and minimum number of single arcs in these digraphs, and their characteristics, is given.     

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$, where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that $\text{Aut}\left(\mathrm{Cay}({\mathrm{Sym}}_n,T_n)\right)$ is the product of the left translation group and a dihedral group ${\mathsf{D}}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $\Gamma$ of $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$ induced by the set $T_n$. In particular, $\Gamma$ is a $2(n?2)$-regular graph whose automorphism group is ${\mathsf{D}}_{n+1},$ $\Gamma$ has as many as $n+1$ maximal cliques of size $2$, and its subgraph $\Gamma \left(V\right)$ whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of ${\mathsf{D}}_{n+1}$ of order $n+1$ with regular Cayley maps on ${\text{Sym}}_n$ is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-$t$-balanced regular Cayley map on ${\text{Sym}}_n$.     

Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices

In this paper, we describe the generation of all nonorientable triangular embeddings of the complete graphs K₁₂ and K₁₃. (The 59 nonisomorphic orientable triangular embeddings of K₁₂ were found in 1996 by Altshuler, Bokowski, and Schuchert, and K₁₃ has no orientable triangular embeddings.) There are 182,200 nonisomorphic nonorientable triangular embeddings for K₁₂, and 243,088,286 for K₁₃. Triangular embeddings of complete graphs are also known as neighborly maps and are a type of twofold triple system. We also use methods of Wilson to provide an upper bound on the number of simple twofold triple systems of order n, and thereby on the number of triangular embeddings of K_n. We mention an application of our results to flexibility of embedded graphs. © 2005 Wiley Periodicals, Inc. J Combin Designs     

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.

     

On graphs with equal domination and 2-domination numbers

Let G be a simple graph, and let p be a positive integer. A subset D⊆V(G) is a p-dominating set of the graph G, if every vertex v∈V(G)-D is adjacent to at least p vertices in D. The p-domination numberγ_p(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination numberγ(G). This definition immediately leads to the inequality γ(G)?γ₂(G).In this paper we present some sufficient as well as some necessary conditions for graphs G with the property that γ₂(G)=γ(G). In particular, we characterize all cactus graphs H with γ₂(H)=γ(H).     

Non-wandering points and the depth for graph maps

Let G be a graph and f:G→G be continuous.Denote by R(f) andΩ(f) the set of recurrent points and the set of non-wandering points of f respectively.LetΩ_0(f) = G andΩ_n(f)=Ω(f|_(Ω_(n-1)(f))) for all n∈N.The minimal m∈NU {∞} such thatΩ_m(f)=Ω_(m 1)(f) is called the depth of f.In this paper,we show thatΩ_2 (f)=(?) and the depth of f is at most 2.Furthermore,we obtain some properties of non-wandering points of f.     

The graphs that have antivoltages using groups of small order

Given a group $\Gamma$ of order at most six, we characterize the graphs that have $\Gamma$-antivoltages and also determine the list of minor-minimal graphs that have no $\Gamma$-antivoltage. Our characterizations yield polynomial-time recognition algorithms for such graphs.     

可迁图的超常边连通度的最优性

图的超常边连通度是图的边连通度概念的推广,对于n阶点可迁或正则边可迁的简单连通图来说,它的h阶超常边连通度λ_h一定存在(1≤h≤n/2)。本文证明了:当d_-正则的n_-阶点可迁简单连通图满足n≥6,d≥4且围长g≥5时,或d_-正则的n_-阶边可迁简单连通图满足n≥6,d≥4且围长g≥4时,对于任何的h:1≤h≤min{g-1,n/2},λ_h达到其最大可能值,即λ_h=hd-2(h-1)。     

The structure of equicontinuous maps

Let be a metric space, and be a continuous map. In this paper we prove that if is compact, and for all , then is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism and a non-expanding map that is pointwise convergent to a fixed point such that is uniformly conjugate to a subsystem of the product map . In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.

     

A census of semisymmetric cubic graphs on up to 768 vertices

A list is given of all semisymmetric (edge- but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3,3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K_1,3 as a normal quotient is investigated in detail. Supported in part by N.Z. Marsden Fund (grant no. UOA 124) and N.Z. Centres of Research Excellence Fund (grant no. UOA 201) Supported in part by “Ministrstvo za šolstvo, znanost in šport Slovenije”, research program no. 101-506. Supported in part by research projects no. Z1-4186-0101 and no. Z1-3124-0101. The fourth author would like to thank the University of Auckland for hospitality during his visit there in 2003.     

Clique coverings and partitions of line graphs

A clique in a graph G is a complete subgraph of G. A clique covering (partition) of G is a collection C of cliques such that each edge of G occurs in at least (exactly) one clique in C. The clique covering (partition) numbercc(G) (cp(G)) of G is the minimum size of a clique covering (partition) of G. This paper gives alternative proofs, using a unified approach, for the results on the clique covering (partition) numbers of line graphs obtained by McGuinness and Rees [On the number of distinct minimal clique partitions and clique covers of a line graph, Discrete Math. 83 (1990) 49-62]. We also employ the proof techniques to give an alternative proof for the De Brujin-Erd?s Theorem.     

Regular Cayley maps for finite abelian groups

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex. Supported in part by the N.Z. Marsden Fund (grant no. UOA0124).