Ted Dobson
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View article: On cubic vertex-transitive graphs of given girth
On cubic vertex-transitive graphs of given girth Open
A set of vertices of a graph is distinguishing if the only automorphism that preserves it is the identity. The minimal size of such sets, if they exist, is the distinguishing cost. The distinguishing costs of vertex transitive cubic graphs…
View article: Classification of Vertex-Transitive Digraphs of Order a Product of Two Distinct Primes via Automorphism Group
Classification of Vertex-Transitive Digraphs of Order a Product of Two Distinct Primes via Automorphism Group Open
In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information c…
View article: AUTOMORPHISM GROUPS OF BICOSET DIGRAPHS
AUTOMORPHISM GROUPS OF BICOSET DIGRAPHS Open
We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X -joins with collections of empty graphs, and show that their automorphism groups can be obt…
View article: Groups with elements of order 8 do not have the DCI property
Groups with elements of order 8 do not have the DCI property Open
Let k be odd, and n an odd multiple of 3. Although this can also be deduced from known results, we provide a new proof that Ck ⋊ C8 and (Cn × C3) ⋊ C8 do not have the Directed Cayley Isomorphism (DCI) property. When k is prime, Ck ⋊ C8 had…
View article: On the BCI Problem
On the BCI Problem Open
Let $G$ be a group. The BCI problem asks whether two Haar graphs of $G$ are isomorphic if and only if they are isomorphic by an element of an explicit list of isomorphisms. We first generalize this problem in a natural way and give a theor…
View article: Recognizing bicoset digraphs which are $X$-joins and automorphism groups of bicoset digraphs
Recognizing bicoset digraphs which are $X$-joins and automorphism groups of bicoset digraphs Open
We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are $X$-joins with collections of empty graphs, and show that their automorphism groups can be ob…
View article: Finding automorphism groups of double coset graphs and Cayley graphs are equivalent
Finding automorphism groups of double coset graphs and Cayley graphs are equivalent Open
It has long been known that a vertex-transitive graph $Γ$ is isomorphic to a double coset graph $\text{Cos}(G,H,S)$ of a transitive group $G\le\text{Aut}(Γ)$, a vertex stabilizer $H\le G$, and some subset $S\subseteq G$. We show that the a…
View article: Groups with elements of order 8 do not have the DCI property
Groups with elements of order 8 do not have the DCI property Open
Let $k$ be odd, and $n$ an odd multiple of $3$. We prove that $C_k \rtimes C_8$ and $(C_n \times C_3)\rtimes C_8$ do not have the Directed Cayley Isomorphism (DCI) property. When $k$ is also prime, $C_k \rtimes C_8$ had previously been pro…
View article: New classes of groups related to algebraic combinatorics with applications to isomorphism problems
New classes of groups related to algebraic combinatorics with applications to isomorphism problems Open
We introduce two refinements of the class of $5/2$-groups, inspired by the classes of automorphism groups of configurations and automorphism groups of unit circulant digraphs. We show that both of these classes have the property that any t…
View article: Towards inductive proofs in algebraic combinatorics
Towards inductive proofs in algebraic combinatorics Open
We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family, …
View article: On automorphisms of Haar graphs of abelian groups
On automorphisms of Haar graphs of abelian groups Open
Let G be a group and S ⊆ G. In this paper, a Haar graph of G with connection set S has vertex set ℤ2 × G and edge set {(0,g)(1,gs) : g ∈ G and s ∈ S}. Haar graphs are then natural bipartite analogues of Cayley digraphs, and are also called…
View article: On Factor-Invariant Graphs With Two Cycles
On Factor-Invariant Graphs With Two Cycles Open
We classify trivalent vertex-transitive graphs whose edge sets have a partition into a 2-factor composed of two cycles and a 1-factor that is invariant under the action of the automorphism group.
View article: Generalized dihedral CI-groups
Generalized dihedral CI-groups Open
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if R is a generalised dihedral group and if R is a CI-group, then for every odd prime p the Sylow p-subgroup of R has order p, or 9. Consequently,…
View article: Generalised dihedral CI-groups
Generalised dihedral CI-groups Open
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$.…
View article: Classification of vertex-transitive digraphs via automorphism group
Classification of vertex-transitive digraphs via automorphism group Open
In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information c…
View article: On the Isomorphism Problem for Cayley Graphs of Abelian Groups whose Sylow Subgroups are Elementary Abelian or Cyclic
On the Isomorphism Problem for Cayley Graphs of Abelian Groups whose Sylow Subgroups are Elementary Abelian or Cyclic Open
We show that if certain arithmetic conditions hold, then the Cayley isomorphism problem for abelian groups, all of whose Sylow subgroups are elementary abelian or cyclic, reduces to the Cayley isomorphism problem for its Sylow subgroups. T…
View article: Some New Groups which are not CI-groups with Respect to Graphs
Some New Groups which are not CI-groups with Respect to Graphs Open
A group $G$ is a CI-group with respect to graphs if two Cayley graphs of $G$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$. We show that an infinite family of groups which include $D_n\times F_{3p}$ are n…
View article: Imprimitive Permutations in Primitive Groups
Imprimitive Permutations in Primitive Groups Open
The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of …