arXiv subjects

Peter J. Cameron

Publications and source records attributed to Peter J. Cameron.

At least 37 records · Page 2Linked to original sources

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.

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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.

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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.

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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.

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arXiv2024

EPPA numbers of graphs

If $G$ is a graph, $A$ and $B$ its induced subgraphs, and $f\colon A\to B$ an isomorphism, we say that $f$ is a \emph{partial automorphism} of $G$. In 1992, Hrushovski proved that graphs have the \emph{extension property for partial automorphisms} (\emph{EPPA}, also called the \emph{Hrushovski property}), that is, for every finite graph $G$ there is a finite graph $H$, an \emph{EPPA-witness} for $G$, such that $G$ is an induced subgraph of $H$ and every partial automorphism of $G$ extends to an automorphism of $H$. The EPPA number of a graph $G$, denoted by $\mathop{\mathrm{eppa}}\nolimits(G)$, is the smallest number of vertices of an EPPA-witness for $G$, and we put $\mathop{\mathrm{eppa}}\nolimits(n) = \max\{\mathop{\mathrm{eppa}}\nolimits(G) : \lvert G\rvert = n\}$. In this note we review the state of the area, prove several lower bounds (in particular, we show that $\mathop{\mathrm{eppa}}\nolimits(n)\geq \frac{2^n}{\sqrt{n}}$, thereby identifying the correct base of the exponential) and pose many open questions. We also briefly discuss EPPA numbers of hypergraphs, directed graphs, and $K_k$-free graphs.

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arXiv2024

On the order sequence of a group

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. M.~Amiri recently proved that the poset has a unique maximal element, corresponding to the cyclic group. We show that the product of orders in a cyclic group of order $n$ is at least $q^{ϕ(n)}$ times as large as the product in any non-cyclic group,where $q$ is the smallest prime divisor of $n$ and $ϕ$ is Euler's function, with a similar result for the sum. The poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.

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arXiv2024

Minimal cover groups

Let $\mathcal{F}$ be a set of finite groups. A finite group $G$ is called an \emph{$\mathcal{F}$-cover} if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. An $\mathcal{F}$-cover is called \emph{minimal} if no proper subgroup of $G$ is an $\mathcal{F}$-cover, and \emph{minimum} if its order is smallest among all $\mathcal{F}$-covers. We prove several results about minimal and minimum $\mathcal{F}$-covers: for example, every minimal cover of a set of $p$-groups (for $p$ prime) is a $p$-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether $\{\mathbb{Z}_q,\mathbb{Z}_r\}$ has finitely many minimal covers, where $q$ and $r$ are distinct primes. Motivated by this, we say that $n$ is a \emph{Cauchy number} if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by $n$, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

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arXiv2024

Super graphs on groups, II

In an earlier paper, the authors considered three types of graphs, and three equivalence relations, defined on a group, viz.\ the power graph, enhanced power graph, and commuting graph, and the relations of equality, conjugacy, and same order; for each choice of a graph type A and an equivalence relation B, there is a graph, the \emph{B superA graph} defined on $G$. The resulting nine graphs (of which eight were shown to be in general distinct) form a two-dimensional hierarchy. In the present paper, we consider these graphs further. We prove universality properties for the conjugacy supergraphs of various types, adding the nilpotent, solvable and enhanced power graphs to the commuting graphs considered in the rest of the paper, and also examine their relation to the invariable generating graph of the group. We also show that supergraphs can be expressed as graph compositions, in the sense of Schwenk, and use this representation to calculate their Wiener index. We illustrate these by computing Wiener index of equality supercommuting and conjugacy supercommuting graphs for dihedral and quaternion groups.

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arXiv2023

Pre-primitive permutation groups

A transitive permutation group $G$ on a finite set $Ω$ is said to be pre-primitive if every $G$-invariant partition of $Ω$ is the orbit partition of a subgroup of $G$. It follows that pre-primitivity and quasiprimitivity are logically independent (there are groups satisfying one but not the other) and their conjunction is equivalent to primitivity. Indeed, part of the motivation for studying pre-primitivity is to investigate the gap between primitivity and quasiprimitivity. We investigate the pre-primitivity of various classes of transitive groups including groups with regular normal subgroups, direct and wreath products, and diagonal groups. In the course of this investigation, we describe all $G$-invariant partitions for various classes of permutation groups $G$. We also look briefly at conditions similarly related to other pairs of conditions, including transitivity and quasiprimitivity, $k$-homogeneity and $k$-transitivity, and primitivity and synchronization.

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arXiv2023

Hypergraphs defined on algebraic structures

There has been a great deal of research on graphs defined on algebraic structures in the last two decades. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective.

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arXiv2023

Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph

The Gruenberg-Kegel graph $Γ(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much recent interest; one of our goals here is to give a survey of some of this material, relating to groups with the same Gruenberg-Kegel graph. However, our main aim is to prove several new results. Among them are the following. - There are infinitely many finite groups with the same Gruenberg-Kegel graph as the Gruenberg-Kegel of a finite group $G$ if and only if there is a finite group $H$ with non-trivial solvable radical such that $Γ(G)=Γ(H)$. - There is a function $F$ on the natural numbers with the property that if a finite $n$-vertex graph whose vertices are labelled by pairwise distinct primes is the Gruenberg-Kegel graph of more than $F(n)$ finite groups, then it is the Gruenberg-Kegel graph of infinitely many finite groups. (The function we give satisfies $F(n)=O(n^7)$, but this is probably not best possible.) - If a finite graph $Γ$ whose vertices are labelled by pairwise distinct primes is the Gruenberg-Kegel graph of only finitely many finite groups, then all such groups are almost simple; moreover, $Γ$ has at least three pairwise non-adjacent vertices, and $2$ is non-adjacent to at least one odd vertex. - Groups whose power graphs, or commuting graphs, are isomorphic have the same Gruenberg-Kegel graph. - The groups ${^2}G_2(27)$ and $E_8(2)$ are uniquely determined by the isomorphism types of their Gruenberg-Kegel graphs. In addition, we consider groups whose Gruenberg-Kegel graph has no edges. These are the groups in which every element has prime power order, and have been studied under the name \emph{EPPO groups}; completing this line of research, we give a complete list of such groups.

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arXiv2023

The number of string C-groups of high rank

If $G$ is a transitive group of degree $n$ having a string C-group of rank $r\geq (n+3)/2$, then $G$ is necessarily the symmetric group $S_n$. We prove that if $n$ is large enough, up to isomorphism and duality, the number of string C-groups of rank $r$ for $S_n$ (with $r\geq (n+3)/2$) is the same as the number of string C-groups of rank $r+1$ for $S_{n+1}$. This result and the tools used in its proof, in particular the rank and degree extension, imply that if one knows the string C-groups of rank $(n+3)/2$ for $S_n$ with $n$ odd, one can construct from them all string C-groups of rank $(n+3)/2+k$ for $S_{n+k}$ for any positive integer $k$. The classification of the string C-groups of rank $r\geq (n+3)/2$ for $S_n$ is thus reduced to classifying string C-groups of rank $r$ for $S_{2r-3}$. A consequence of this result is the complete classification of all string C-groups of $S_n$ with rank $n-κ$ for $κ\in\{1,\ldots,6\}$, when $n\geq 2κ+3$, which extends previously known results. The number of string C-groups of rank $n-κ$, with $n\geq 2κ+3$, of this classification gives the following sequence of integers indexed by $κ$ and starting at $κ= 1$: $$(1,1,7,9,35,48)$$ This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It will be available as sequence number A359367.

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arXiv2022

Acyclic orientations and poly-Bernoulli numbers

The main contribution of this paper is a formula for the number of acyclic orientations of a complete bipartite, $K_{n_1,n_2},$ revealing that it is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$ introduced in 1997 by Kaneko. We also give a simple bijective identification of acyclic orientations and lonesum matrices, confirming a 2008 result of Brewbaker, and show that the poly-Bernoulli numbers behave concavely, as well being symmetric. A second goal is to explore the behaviour of more general complete $r$-partite graphs in the space of the number of acyclic orientations. We prove that the number of acyclic orientations of complete bipartite graphs on $n$ vertices is a unimodular concave function maximised by the Turán graph $T(2,n).$ For tripartite graphs, we derive an explicit formula for its number of its acyclic orientations, for the specific case with a single vertex partition. For more complex complete $r$-partite graphs an algorithmic approach is suggested, taking advantage of the relationship with graph colouring through the result of Stanley, namely the value of the graph colouring polynomial at the value -1. An underlying theme is the exploration of the space of the number of acyclic orientations of graphs. We present various theoretical and computational results, with additional, conjectures for both complete $r$-partite and general graphs to guide future research.

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arXiv2022

On Difference of Enhanced Power Graph and Power Graph of a Finite Group

The difference graph $D(G)$ of a finite group $G$ is the difference of enhanced power graph of $G$ and power graph of $G$, with all isolated vertices are removed. In this paper we study the connectedness and perfectness of $D(G)$ with respect to various properties of the underlying group $G$. We also find several connection between the difference graph of $G$ and the Gruenberg-Kegel graph of $G$.

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arXiv2022

Number of spanning trees containing a given forest

We consider all spanning trees of a complete simple graph $Γ$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $τ(F)$, doesn't depend on the structure of $F$ and is completely determined by the number of vertices $q_i \, (i=1, ..., m)$ in each connected component of $F$. Specifically, $τ(F) = q_1 q_2 \cdots q_m n^{m-2}$.

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arXiv2022

Component graphs of vector spaces and zero-divisor graphs of ordered sets

In this paper, nonzero component graphs and nonzero component union graphs of finite dimensional vector space are studied using the zero-divisor graph of specially constructed 0-1-distributive lattice and the zero-divisor graph of rings. Further, we define an equivalence relation on nonzero component graphs and nonzero component union graphs to deduce that these graphs are the graph join of zero-divisor graphs of Boolean algebras and complete graphs. In the last section, we characterize the perfect and chordal nonzero component graphs and nonzero component union graphs.

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arXiv2022

Association schemes with given stratum dimensions: on a paper of Peter M. Neumann

In January 1969, Peter M. Neumann wrote a paper entitled "Primitive permutation groups of degree 3p". The main theorem placed restrictions on the parameters of a primitive but not 2-transitive permutation group of degree three times a prime. The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree. However, there are further reasons for being interested in this paper. First, it was written at a time when combinatorial techniques were being introduced into the theory of finite permutation groups, and the paper gives a very good summary and application of these techniques. Second, like its predecessor by Helmut Wielandt on primitive groups of degree 2p, it can be re-interpreted as a combinatorial result concerning association schemes whose common eigenspaces have dimensions of a rather limited form. This result uses neither the primality of p nor the existence of a permutation group related to the combinatorial structure. We extract these results and give details of the related combinatorics.

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arXiv2022

Generalized non-coprime graphs of groups

Let G be a finite group with identity e and H \neq \{e\} be a subgroup of G. The generalized non-coprime graph GAmma_{G,H} of G with respect to H is the simple undirected graph with G - \{e \}\) as the vertex set and two distinct vertices a and b are adjacent if and only if \gcd(|a|,|b|) \neq 1 and either a \in H or b \in H, where |a| is the order of a\in G. In this paper, we study certain graph theoretical properties of generalized non-coprime graphs of finite groups, concentrating on cyclic groups. More specifically, we obtain necessary and sufficient conditions for the generalized non-coprime graph of a cyclic group to be in the class of stars, paths, cycles, triangle-free, complete bipartite, complete, unicycle, split, claw-free, chordal or perfect graphs. Then we show that widening the class of groups to all finite nilpotent groups gives us no new graphs, but we give as an example of contrasting behaviour the class of EPPO groups (those in which all elements have prime power order). We conclude with a connection to the Gruenberg--Kegel graph.

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