Connected Cayley Graphs Research Articles

On Automorphisms of Direct Products of Cayley Graphs on Abelian Groups

Let $X$ and $Y$ be connected Cayley graphs on abelian groups, such that no two distinct vertices of $X$ have exactly the same neighbours, and the same is true about $Y$. We show that if the number of vertices of $X$ is relatively prime to the number of vertices of $Y$, then the direct product $X \times Y$ has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $Y$). This was not previously known even in the special case where $Y = K_2$ has only two vertices. The proof of this special case is short and elementary. The general case follows from the special case by standard arguments.

Characterizing a vertex‐transitive graph by a large ball

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgakopoulos. Answering a question of Cornulier, we also construct a continuum of non pairwise isometric large-scale simply connected locally finite vertex-transitive graphs. This question was motivated by the fact that large-scale simply connected Cayley graphs are precisely Cayley graphs of finitely presented groups and therefore have countably many isometric classes.

Non-expander Cayley Graphs of Simple Groups

For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.

An Intermediate Value Theorem for the Decycling Numbers of Toeplitz Graphs

In this paper we obtain an intermediate value theorem for the decycling numbers of Toeplitz graphs: if $$n \ge 3$$n?3, and $$0 \le r \le n - 2$$0≤r≤n-2, then there exists a Toeplitz graph of order $$n$$n with decycling number $$r$$r. We also prove that the decycling numbers of connected Cayley graphs of order $$n$$n satisfy the intermediate value property if and only if $$n = 4$$n=4 or $$6$$6.

Hamilton paths in Cayley graphs on Coxeter groups: I

We consider several families of Cayley graphs on the finite Coxeter groups A n , B n , and D n with regard to the problem of whether they are Hamilton-laceable or Hamilton-connected. It is known that every connected bipartite Cayley graph on A n , n ≥ 2 , whose connection set contains only transpositions and has valency at least three is Hamilton-laceable. We obtain analogous results for connected bipartite Cayley graphs on B n , and for connected Cayley graphs on D n . Non-bipartite examples arise for the latter family.

Hamilton paths in generalized Petersen graphs

Alspach and Qin proved that connected Cayley graphs of Hamiltonian groups (all subgroups are normal) are either Hamilton-connected (every pair of vertices is joined by a Hamilton path), or are bipartite and Hamilton-laceable (every pair on opposite sides of the bipartition are joined by a Hamilton path). Their proof made use of Hamilton-connectedness of certain Generalized Petersen graphs.In this work, we extend (and make a small correction to) the results of Alspach and Liu on Hamilton paths in generalized Petersen graphs. Alspach and Liu showed that, for k∈{1,2,3} and gcd(n,k)=1, P(n,k) is either Hamilton-connected or bipartite and Hamilton-laceable, as long as (n,k)≠(6r+5,2) or (5,3). For k=2, we consider the remaining cases for n and completely determine which pairs of vertices in P(n,2) are joined by Hamilton paths. However, the main point is to show that, for each k, it is a finite problem to determine, for all n, which pairs of vertices in P(n,k) are the ends of a Hamilton path.

Pancyclicity and Cayley Graphs on Abelian Groups

AbstractWe prove that connected Cayley graphs of valency at least 3 on abelian groups are even edge‐pancyclic and have cycles of every possible odd length bigger than or equal to the odd girth.

Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

AbstractIn this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ isn‐HC‐extendableif it contains a path of lengthnand if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ isweakly n‐HP‐extendableif it contains a path of lengthnand if every such path is contained in some Hamilton path of Γ. Moreover, Γ isstrongly n‐HP‐extendableif it contains a path of lengthnand if for every such pathPthere is a Hamilton path of Γ starting withP. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2‐HC‐extendable and a complete classification of 3‐HC‐extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4‐HP‐extendable. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory

On the extendability of certain semi-Cayley graphs of finite abelian groups

A connected graph Γ with at least 2 n + 2 vertices is said to be n -extendable if every matching of size n in Γ can be extended to a perfect matching. The aim of this paper is to study the 1-extendability and 2-extendability of certain semi-Cayley graphs of finite abelian groups, and the classification of connected 2-extendable semi-Cayley graphs of finite abelian groups is given. Thus the 1-extendability and 2-extendability of Cayley graphs of non-abelian groups which can be realized as such semi-Cayley graphs of abelian groups can be deduced. In particular, the 1-extendability and 2-extendability of connected Cayley graphs of generalized dicyclic groups and generalized dihedral groups are characterized.

Hamilton paths in Cayley graphs on generalized dihedral groups

We investigate the existence of Hamilton paths in connected Cayley graphs on generalized dihedral groups. In particular, we show that a connected Cayley graph of valency at least three on a generalized dihedral group, whose order is divisible by four, is Hamilton-connected, unless it is bipartite, in which case it is Hamilton-laceable.

On extendability of Cayley graphs

A connected graph ? of even order is n-extendable, if it contains a matching of size n and if every such matching is contained in a perfect matching of ?. Furthermore, a connected graph ? of odd order is n1/2-extendable, if for every vertex v of ? the graph ? - v is n-extendable. It is proved that every connected Cayley graph of an abelian group of odd order which is not a cycle is 1 1/2-extendable. This result is then used to classify 2-extendable connected Cayley graphs of generalized dihedral groups.

Normality of Tetravalent Cayley Graphs of Odd Prime–cube Order and Its Application

Let p be an odd prime. In this paper we prove that all tetravalent connected Cayley graphs of order p3 are normal. As an application, a classification of tetravalent symmetric graphs of odd prime–cube order is given.

Hamiltonian decomposition of Cayley graphs of ordersp 2 andpq

In this paper, it is proved that any connected Cayley graph on an abelian group of order pq orp2 has a hamiltonian decomposition, wherep andq are odd primes. This result answers partially a conjecture of Alspach concerning hamiltonian decomposition of 2k-regular connected Cayley graphs on abelian groups.

On Isomorphisms of Connected Cayley Graphs, II

A Cayley graph or digraph Cay(G, S) of a finite groupGis called aCI-graphofGif, for anyT⊂G, Cay(G, S)≅Cay(G, T) if and only ifSσ=Tfor someσ∈Aut(G). We study the problem of determining which Cayley graphs and digraphs for a given group are CI-graphs. A finite groupGis called aconnected m-DCI-group(orconnected m-CI-group) if all connected Cayley digraphs (or connected Cayley graphs, respectively) ofGof (out)-valency at most m are CI-graphs. For a groupG, letp(G) be the smallest prime divisor of |G|. It was previously shown that all finite groupsGare connected (p(G)−1)-DCI-groups and all finite groupsGare connected 2(p(G)−1)-CI-groups. In this paper, for every primep, we construct infinitely many finite groupsGsuch thatp(G)=pandGis neither a connectedp-DCI-group nor a connected 2p-CI-group, which provides solutions for several open problems in this area.

On isomorphisms of connected Cayley graphs, III

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for m ≤ p(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

Isomorphisms of Cayley digraphs of Abelian groups

For a finite group G and a subset S of G with 1 ∉ S, the Cayley graph Cay(G, S) is the digraph with vertex set G such that (x, y) is an arc if and only if yx−1 ∈ S. The Cayley graph Cay(G, S) is called a CI-graph if, for any T ⊂ G, whenever Cay (G, S) ≅ Cay(G, T) there is an element a σ ∈ Aut(G) such that Sσ = T. For a positive integer m, G is called an m-DCI-group if all Cayley graphs of G of valency at most m are CI-graphs; G is called a connected m-DCI-group if all connected Cayley graphs of G of valency at most m are CI-graphs. The problem of determining Abelian m-DCI-groups is a long-standing open problem. It is known from previous work that all Abelian m-DCI-groups lie in an explicitly determined class of Abelian groups. First we reduce the problem of determining Abelian m-DCI-groups to the problem of determining whether every subgroup of a member of is a connected m-DCI-group. Then (for a finite group G, letting p be the least prime divisor of |G|,) we completely classify Abelian connected (p + 1)-DCI-groups G, and as a corollary, we completely classify Abelian m-DCI-groups G for m ≤ p + 1. This gives many earlier results when p = 2.

Isomorphisms of Connected Cayley Digraphs

Let G be a finite group and let Cay(\(\)) be a Cayley graph of G. The graph Cay(\(\)) is called a CI-graph of G if, for any \(\) for some \(\)Aut(G) only when Cay\(\)Cay(\(\)). In this paper, we study the isomorphism problem of connected Cayley graphs: to determine the groups G (or the types of Cayley graphs for a given group G) for which all connected Cayley graphs for G are CI-graphs.

On adjacency-transitive graphs

Let Γ be a finite simple undirected graph. An automorphism σ ∈ Aut Γ is an adjacency automorphism of Γ if dist( x, σ( x)) ⩽ 1 for every vertex x ∈ Γ. A graph Γ is adjacency-transitive if for every pair of vertices x, y ∈ V( Γ) there exists a sequence of adjacency automorphisms σ 1, σ 2, …, σ k ∈ Aut Γ such that σ 1 σ 2 ··· σ k ( x) = y. Examples of such graphs include certain classes of connected Cayley graphs, but not all of them. Some basic properties and examples of adjacency-transitive graphs are given and those of valency 3 and 4 are classified.

On isomorphisms of connected Cayley graphs

Let G be a finite group and Cay( G, S) the Cayley graph of G with respect to S. A subset S is called a CI-subset if, for any T ⊂ G, Cay( G, S) ≅ Cay( G, T) implies S α = T for some α ∈ Aut( G). In this paper, we investigate the finite groups G in which every subset S with size at most m and (S⋋ = G is a CI-subset where m is a positive integer. As a corollary, we classify symmetric graphs of order p 3 and of valency 2 p where p is a prime.

Pseudo-cartesian product and hamiltonian decompositions of Cayley graphs on abelian groups

Alspach has conjectured that any 2 k-regular connected Cayley graph cay( A, S) on a finite abelian group A can be decomposed into k hamiltonian cycles. In this paper we generalize a result by Kotzig that the cartesian product of any two cycles can be decomposed into two hamiltonian cycles and show that any pseudo-cartesian product of two cycles can be decomposed into two hamiltonian cycles. By applying that result we first give an alternative proof for the main result in (Bermond et al., 1989), including the missing cases, and then we show that the conjecture is true for most 6-regular connected Cayley graphs on abelian groups of odd order and for some 6-regular connected Cayley graphs on abelian groups of even order.