arXiv subjects

Peter J. Cameron

Publications and source records attributed to Peter J. Cameron.

At least 19 recordsLinked to original sources

arXiv2026

Complete Mappings of Semigroups

A complete mapping of a semigroup $S$ is a bijection $\alpha\colon S\to S$ such that the map $\theta\colon S\to S$ defined by $x\theta=x\cdot x\alpha$ is also a bijection. Equivalently, it determines a transversal of the multiplication table of $S$. Complete mappings connect group theory, Latin squares, and cryptography, and their existence for finite groups was characterized by the resolution of the Hall--Paige conjecture. In this paper, we develop the corresponding theory for finite semigroups. We prove that every finite semigroup admitting a complete mapping is regular and that the problem reduces to principal factors. We classify the existence of a complete mapping in Rees matrix semigroups without zero, give a Hall-type criterion for Rees $0$-matrix semigroups over groups with complete mappings, and prove sufficient conditions for Rees $0$-matrix semigroups whose maximal subgroups do not have complete mappings. As the main application of the Rees $0$-matrix analysis, we show that $T_n$ has a complete mapping if and only if $n=1$ or $n\geq 4$. Equivalently, $T_n$ has a complete mapping if and only if the same holds for $S_n$. We prove that the full linear monoid of a finite-dimensional vector space has a complete mapping except in dimension $1$ over a field of odd order and in dimension $2$ over $\mathbb F_2$. We also prove that the partition monoid $\mathcal P_n$ has a complete mapping if and only if $n=1$ or $n\ge4$, and that every finite aperiodic regular $*$-semigroup has a complete mapping. As a consequence, the planar partition, Motzkin and Jones monoids have complete mappings. The paper concludes with open problems.

math.GR↗

arXiv2026

On modular balanced partition designs

Let $X$ be a finite set of integers with cardinality $\nu = \kappa \lambda$. A \emph{modular balanced partition design} is a triplet $(X, \mathcal{A}, \mathcal{B})$ satisfying the following conditions: \begin{itemize} \item $\mathcal{A}$ is a partition of $X$ into $\kappa$ blocks of size $\lambda$, such that every element of $X$ appears in exactly one block. If $\mathcal{A} = \{A_1, A_2, \ldots, A_{\kappa}\}$, then $\sum_{a\in A_i} a \equiv i \lambda \pmod{\nu}$, for each $i=1,2,\ldots,\kappa$ \item $\mathcal{B}$ is a partition of $X$ into $\lambda$ blocks of size $\kappa$, such that every element of $X$ appears in exactly one block. If $\mathcal{B} = \{B_1, B_2, \ldots, B_{\lambda}\}$, then $\sum_{b\in B_j} b \equiv j \kappa \pmod{\nu}$, for each $j=1,2,\ldots,\lambda$ \item $A_i \cap B_j$ has exactly one element, for any $A_i \in \mathcal{A}$, and $B_j \in \mathcal{B}$. \itemize} We prove the necessary conditions for the existence of a modular balanced partition design. Moreover, we investigate and identify a relationship between a modular balanced partition design and a subgroup magic rectangle. Then by using affine automorphisms of an Abelian group, we prove the existence of non-isomorphic modular balanced partition designs. Finally, we provide a method to construct a transversal design via a modular balanced partition design.

math.CO↗

arXiv2026

Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs

An equitable partition of a graph $\Ga$ is a partition $\{V_1, \ldots, V_m\}$ of its vertex set such that for each pair $i, j$ all vertices in $V_i$ have the same number of neighbours in $V_j$. When $m=2$, $V_1$ is called an $(a, b)$-perfect set in $\Ga$, where $a$ is the number of neighbours in $V_1$ of each vertex in $V_1$, and $b$ is the number of neighbours in $V_1$ of each vertex in $V_2$. In this paper we first derive general necessary conditions for a regular graph to admit two equitable partitions. As a corollary we obtain necessary conditions for the existence of an $(a,b)$-perfect set in a regular graph in terms of an arbitrary equitable partition. With the help of these results we then obtain necessary conditions for the existence of an $(a,b)$-perfect set in a normal Cayley graph in terms of the irreducible characters of the underlying group.

math.CO↗

arXiv2026

Filters, topologies, the Rado graph and the Urysohn space

My work with Anatoly Vershik concerned automorphism groups of the Rado graph and homeomorphism groups of the Urysohn space. This paper contains some further thoughts on these issues, together with connections to topologies and filters on countable sets.

math.GR↗

arXiv2026

Around homogeneity

Forty-five years ago, a young researcher in finite permutation group theory encountered a paper by Robert Woodrow. The homogeneous triangle-free graph Woodrow described there seemed to be an infinite analogue of the Higman--Sims graph which had played an important role in the researcher's thesis. The encounter changed the course of the researcher's career. This paper is the story of that event and its aftermath. The final section of the paper suggests that Fra\"{\i}ss\'e classes of rigid structures are a potentially interesting generalisation of Ramsey classes.

math.CO↗

arXiv2026

Graphs defined on algebras

There has been a great deal of attention recently to graphs whose vertex set is a group, defined using the group structure. (The commuting graph, where two elements are joined if they commute, is the oldest and most famous example.) The purpose of this paper is to investigate extending the definitions of such graphs to general algebras (in the sense of universal algebra). It seems unlikely that such a definition can be made for the commuting graph, or for various others such as the nilpotency and Engel graphs. However, for graphs whose definition depends on the notion of subgroup or subalgebra generated by a subset, the existing definitions work without change. These graphs include several well-studied examples: the power graph, enhanced power graph, generating graph, independence graph, and rank graph. In these cases, some results about groups extend to arbitrary algebras unchanged, but others require specific properties of groups, and pose a challenge to researchers. In the next two sections, I will describe some extensions to directed graphs (the directed power graph and the endomorphism digraph) and to simplicial complexes (the independence and strong independence complexes). The final section gives explicit descriptions of all of these objects for independence algebras.

math.CO↗

arXiv2025

A footnote to the KPT theorem in structural Ramsey theory

The celebrated theorem of Kechris, Pestov and Todor\v{c}evi\'c connecting structural Ramsey theory with topological dynamics has as a consequence that the Fra\"{\i}ss\'e limit of a Ramsey class of non-trivial finite relational structures has a reduct which is a total order; this implies an earlier result of Ne\v{s}et\v{r}il, according to which the structures in such a class are rigid (have trivial automorphism groups). In this paper, we give an alternative proof of this fact. If $\mathcal{C}$ is a Fra\"{\i}ss\'e class of rigid structures over a finite relational language, then either the Fra\"{\i}ss\'e limit of $\mathcal{C}$ has a reduct which is a total order, or there is an explicit failure of the Ramsey property involving a pair $(A,B)$ of structures in $\mathcal{C}$ with $|A|=2$.

math.LO↗

arXiv2025

On the difference of the intersection power graph and the power graph of a finite group

The difference graph D(G) of a finite group G is the graph obtained by taking the (edge) difference of the intersection power graph and the power graph of G, and subsequently removing all isolated vertices. In this paper, we give a number of results about the difference graph. We examine groups whose power graph and intersection power graph coincide. In addition, we make some observations on isolated vertices in difference graphs. We study the connectedness and perfectness of difference graph with respect to various properties of the underlying group G. Furthermore, we investigate the operation of twin reduction on graphs, a technique that yields smaller graphs which may be easier to analyze.

math.CO↗

arXiv2025

Perfect codes in Cayley graphs of abelian groups

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such that every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

math.CO↗

arXiv2025

A new look at twin reduction

Twin reduction defines an equivalence relation on the vertex set of a graph. I give a characterisation of this equivalence relation. A consequence is a structure theorem for the automorphism group of the graph.

math.CO↗

arXiv2025

Hall's marriage theorem

In 1935, Philip Hall published what is often referred to as ``Hall's marriage theorem'' in a short paper (P.~Hall, On Representatives of Subsets, \textit{J. Lond. Math. Soc.} (1) \textbf{10} (1935), no.1, 26--30.) This paper has been very influential. I state the theorem and outline Hall's proof, together with some equivalent (or stronger) earlier results, and proceed to discuss some the many directions in combinatorics and beyond which this theorem has influenced.

math.CO↗

arXiv2024

On the metric dimension of the character degree graph of a solvable group

Let $G$ be a finite solvable group and let $Δ(G)$ be the character degree graph of $G$. In this paper, we obtain the metric dimension of certain character degree graphs. Specifically, we calculate the metric dimension for a regular character degree graph, a character degree graph with a diameter of $2$ that is not a block, a character degree graph with a diameter of $3$ that also has a cut vertex and a character degree graph with Fitting height $2.$ We also consider two related parameters, base size and adjacency dimension, and their relation to metric dimension for character degree graphs of solvable groups.

math.GR↗

arXiv2024

Simplicial complexes defined on groups

This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the group, and in many cases have a relation to familiar graphs on the group. The ones which seem to reach deepest into the graph structure are two forms of independence complex, and some results on the class of groups for which these two complexes coincide are given. Other examples are treated more briefly.

math.CO↗

arXiv2024

Permutation groups, partition lattices and block structures

Let $G$ be a transitive permutation group on $\Omega$. The $G$-invariant partitions form a sublattice of the lattice of all partitions of $\Omega$, having the further property that all its elements are uniform (that is, have all parts of the same size). If, in addition, all the equivalence relations defining the partitions commute, then the relations form an \emph{orthogonal block structure}, a concept from statistics; in this case the lattice is modular. If it is distributive, then we have a \emph{poset block structure}, whose automorphism group is a \emph{generalised wreath product}. We examine permutation groups with these properties, which we call the \emph{OB property} and \emph{PB property} respectively, and in particular investigate when direct and wreath products of groups with these properties also have these properties. A famous theorem on permutation groups asserts that a transitive imprimitive group $G$ is embeddable in the wreath product of two factors obtained from the group (the group induced on a block by its setwise stabiliser, and the group induced on the set of blocks by~$G$). We extend this theorem to groups with the PB property, embeddng them into generalised wreath products. We show that the map from posets to generalised wreath products preserves intersections and inclusions. We have included background and historical material on these concepts.

math.GR↗

arXiv2024

Problems from BCC30

These problems were mostly presented at the problem session at the 30th British Combinatorial Conference at Queen Mary University of London on 4 July 2024. Some were contributed later by conference participants. Thank you to all the contributors. The problems are given here in alphabetical order of presenter. If no originator is given, I assume that the presenter is the originator. Please send corrections to me (\texttt{pjc20@st-andrews.ac.uk}). Solutions should be sent to the presenter; I would appreciate a copy too.

math.CO↗

arXiv2024

Engel and co-Engel graphs of finite groups

Let $G$ be a group. Associate a directed graph $\vec{E}(G)$ (called the Engel digraph of $G$) with $G$ whose vertex set is $G$, with an arc $(x,y)$ if $[y, {}_k x]=1$ for some positive integer $k$, where $[y,{}_kx]$ is the iterated commutator $[y,x,x,\ldots,x]$, with $k$ terms $x$ in the expression. From this we define the Engel graph $E(G)$ by ignoring directions; the co-Engel graph $E_c(G)$ is its complement. The co-Engel graph, under the name ``Engel graph'', was introduced by Abdollahi. However, the name we use is more natural. We begin with some general results about the Engel digraph and graph, before turning our attention to the co-Engel graph. Among other things, we show that the undirected Engel graph does not determine the directed version up to isomorphism, though counterexamples seem to be fairly rare: there are just two orders less than $100$ for which this happens. We also prove a universality theorem: every finite digraph is an induced sub-digraph of the Engel digraph of a finite group. The isolated vertices of $E_c(G)$ form the Fitting subgroup $F(G)$ of $G$. In this paper, we realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups $G$ induced by $G \setminus F(G)$. We write $E_c^-(G)$ to denote the subgraph of $E_c(G)$ induced by $G \setminus F(G)$. We also compute genus, various spectra, energies and Zagreb indices of $E_c^-(G)$ for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number of $E_c^-(G)$ is at most $4$ and $E_c^-(G)$ is toroidal or projective. Further, we show that $E_c^-(G)$ is ALQ-integral and satisfies the E-LE conjecture and the Hansen-Vuki{\v{c}}evi{\'c} conjecture for the groups considered in this paper.

math.GR↗

arXiv2024

Main functions and the spectrum of super graphs

Let A be a graph type and B an equivalence relation on a group $G$. Let $[g]$ be the equivalence class of $g$ with respect to the equivalence relation B. The B superA graph of $G$ is an undirected graph whose vertex set is $G$ and two distinct vertices $g, h \in G$ are adjacent if $[g] = [h]$ or there exist $x \in [g]$ and $y \in [h]$ such that $x$ and $y$ are adjacent in the A graph of $G$. In this paper, we compute spectrum of equality/conjugacy supercommuting graphs of dihedral/dicyclic groups and show that these graphs are not integral.

math.CO↗

arXiv2024

Regular bipartite multigraphs have many (but not too many) symmetries

Let $k$ and $l$ be integers, both at least 2. A $(k,l)$-bipartite graph is an $l$-regular bipartite multigraph with coloured bipartite sets of size $k$. Define $\chi(k,l)$ and $\mu(k,l)$ to be the minimum and maximum order of automorphism groups of $(k,l)$-bipartite graphs, respectively. We determine $\chi(k,l)$ and $\mu(k,l)$ for $k\geq 8$, and analyse the generic situation when $k$ is fixed and $l$ is large. In particular, we show that almost all such graphs have automorphism groups which fix the vertices pointwise and have order far less than $\mu(k,l)$. These graphs are intimately connected with both contingency tables with uniform margins and uniform set partitions; we examine the uniform distribution on the set of $k\times k$ contingency tables with uniform margin $l$, showing that with high probability all entries stray far from the mean. We also show that the symmetric group acting on uniform set partitions is non-synchronizing.

math.CO↗