Joy Morris
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View article: Detecting Graphical and Digraphical Regular Representations in Groups of Squarefree Order
Detecting Graphical and Digraphical Regular Representations in Groups of Squarefree Order Open
A necessary condition for a Cayley digraph Cay$(R,S)$ to be a regular representation is that there are no non-trivial group automorphisms of $R$ that fix $S$ setwise. A group is DRR-detecting or GRR-detecting if this condition is also suff…
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: Haar graphical representations of finite groups and an application to poset representations
Haar graphical representations of finite groups and an application to poset representations Open
Let R be a group and let S be a subset of R. The Haar graph Haar(R,S) of R with connection set S is the graph having vertex set R×{−1,1}, where two distinct vertices (x,−1) and (y,1) are declared to be adjacent if and only if yx−1∈S. The n…
View article: A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups
A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups Open
In this paper we introduce a new way of measuring the robustness of Erdős--Ko--Rado (EKR) Theorems on permutation groups. EKR-type results can be viewed as results about the independence numbers of certain corresponding graphs, namely the …
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: Robustness of Erdős--Ko--Rado theorems on permutations and perfect matchings
Robustness of Erdős--Ko--Rado theorems on permutations and perfect matchings Open
The Erdős--Ko--Rado (EKR) theorem and its generalizations can be viewed as classifications of maximum independent sets in appropriately defined families of graphs, such as the Kneser graph $K(n,k)$. In this paper, we investigate the indepe…
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: Haar graphical representations of finite groups and an application to poset representations
Haar graphical representations of finite groups and an application to poset representations Open
Let $R$ be a group and let $S$ be a subset of $R$. The Haar graph $\mathrm{Haar}(R,S)$ of $R$ with connection set $S$ is the graph having vertex set $R\times\{-1,1\}$, where two distinct vertices $(x,-1)$ and $(y,1)$ are declared to be adj…
View article: Z_3^8 is not a CI-group
Z_3^8 is not a CI-group Open
A Cayley graph Cay(G, S) has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay(G, T), there is a group automorphism α of G such that Sα = T. The DCI (Directed Cayley Isomorphism) property is defined analogously on digr…
View article: Dihedral groups of order $2pq$ or $2pqr$ are DCI
Dihedral groups of order $2pq$ or $2pqr$ are DCI Open
A group has the (D)CI ((Directed) Cayley Isomorphism) property, or more commonly is a (D)CI group, if any two Cayley (di)graphs on the group are isomorphic via a group automorphism. That is, $G$ is a (D)CI group if whenever $\rm{Cay}(G,S)\…
View article: Detecting Graphical and Digraphical Regular Representations in groups of squarefree order
Detecting Graphical and Digraphical Regular Representations in groups of squarefree order Open
A necessary condition for a Cayley digraph Cay$(R,S)$ to be a regular representation is that there are no non-trivial group automorphisms of $R$ that fix $S$ setwise. A group is DRR-detecting or GRR-detecting if this condition is also suff…
View article: $\mathbb Z_3^8$ is not a CI-group
$\mathbb Z_3^8$ is not a CI-group Open
A Cayley graph Cay$(G;S)$ has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay$(G;T)$, there is a group automorphism $α$ of $G$ such that $S^α=T$. The DCI (Directed Cayley Isomorphism) property is defined analogously …
View article: Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs
Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs Open
A finite group $G$ is a "non-DCI group" if there exist subsets $S_1$ and $S_2$ of $G$, such that the associated Cayley digraphs $C\overrightarrow{ay}(G;S_1)$ and $C\overrightarrow{ay}(G;S_2)$ are isomorphic, but no automorphism of $G$ carr…
View article: Cayley graphs on abelian and generalized dihedral groups
Cayley graphs on abelian and generalized dihedral groups Open
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. In this paper we give conditions for when a Cayley graph on an abelian group can be represented as a C…
View article: Induced forests in some distance-regular graphs
Induced forests in some distance-regular graphs Open
In this article, we study the order and structure of the largest induced forests in some families of graphs. First we prove a variation of the ratio bound that gives an upper bound on the order of the largest induced forest in a graph. Nex…
View article: Cayley graphs on non-isomorphic groups
Cayley graphs on non-isomorphic groups Open
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. In this paper we give conditions for when a Cayley graph on an abelian group can be represented as a C…
View article: On the asymptotic enumeration of Cayley graphs
On the asymptotic enumeration of Cayley graphs Open
In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representatio…
View article: On generalised Petersen graphs of girth 7 that have cop number 4
On generalised Petersen graphs of girth 7 that have cop number 4 Open
We show that if $n=7k/i$ with $i \in \{1,2,3\}$ then the cop number of the generalised Petersen graph $GP(n,k)$ is $4$, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any …
View article: Groups for which it is easy to detect graphical regular representations
Groups for which it is easy to detect graphical regular representations Open
We say that a finite group G is "DRR-detecting" if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontri…
View article: Two families of graphs that are Cayley on nonisomorphic groups
Two families of graphs that are Cayley on nonisomorphic groups Open
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the gr…
View article: Cayley graphs of more than one abelian group
Cayley graphs of more than one abelian group Open
We show that for certain integers $n$, the problem of whether or not a Cayley digraph $Γ$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a natu…
View article: Two new families of non-CCA groups
Two new families of non-CCA groups Open
We determine two new infinite families of Cayley graphs that admit colour-preserving automorphisms that do not come from the group action. By definition, this means that these Cayley graphs fail to have the CCA (Cayley Colour Automorphism)…
View article: Two families of graphs that are Cayley on nonisomorphic groups
Two families of graphs that are Cayley on nonisomorphic groups Open
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the gr…
View article: On the base size of the symmetric and the alternating group acting on partitions
On the base size of the symmetric and the alternating group acting on partitions Open
Given three positive integers n,a,b with n=ab, we determine the base size of the symmetric group and of the alternating group of degree n in their action on the set of partitions into b parts having cardinality a.
View article: Asymptotic enumeration of Cayley digraphs
Asymptotic enumeration of Cayley digraphs Open
In this paper we show that almost all Cayley digraphs have automorphism group as small as possible; that is, they are digraphical regular representations (DRRs). More precisely, we show that as r tends to infinity, for every finite group R…
View article: A finite simple group is CCA if and only if it has no element of order four
A finite simple group is CCA if and only if it has no element of order four Open
A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected C…