A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions

AbstractWe show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also prove that the graphs arising this way are hamiltonian provided that their valency is at least six

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Surfaces with given Automorphism Group

Frucht showed that, for any finite group $G$, there exists a cubic graph such that its automorphism group is isomorphic to $G$. For groups generated by two elements we simplify his construction to a graph with fewer nodes. In the general case, we address an oversight in Frucht's construction. We prove the existence of cycle double covers of the resulting graphs, leading to simplicial surfaces with given automorphism group. For almost all finite non-abelian simple groups we give alternative constructions based on graphic regular representations. In the general cases $C_n,D_n,A_5$ for $n\geq 4$, we provide alternative constructions of simplicial spheres. Furthermore, we embed these surfaces into the Euclidean 3-Space with equilateral triangles such that the automorphism group of the surface and the symmetry group of the corresponding polyhedron in $\mathrm{O}(3)$ are isomorphic

Between primitive and 2-transitive : synchronization and its friends

The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.PostprintPeer reviewe

The diagonal graph

According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a much larger class than those occurring in this theorem. For any positive integer m and group G (finite or infinite), there is a diagonal semilattice, a sub-semilattice of the lattice of partitions of a set Ω, whose automorphism group is the corresponding diagonal group. Moreover, there is a graph (the diagonal graph), bearing much the same relation to the diagonal semilattice and group as the Hamming graph does to the Cartesian lattice and the wreath product of symmetric groups. Our purpose here, after a brief introduction to this semilattice and graph, is to establish some properties of this graph. The diagonal graph ΓD(G,m) is a Cayley graph for the group Gm, and so is vertex-transitive. We establish its clique number in general and its chromatic number in most cases, with a conjecture about the chromatic number in the remaining cases. We compute the spectrum of the adjacency matrix of the graph, using a calculation of the Möbius function of the diagonal semilattice. We also compute some other graph parameters and symmetry properties of the graph. We believe that this family of graphs will play a significant role in algebraic graph theory.PostprintPeer reviewe

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Parameterized (Modular) Counting and Cayley Graph Expanders

We study the problem #EdgeSub(?) of counting k-edge subgraphs satisfying a given graph property ? in a large host graph G. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of p-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field ?_p which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of #EdgeSub(?) for minor-closed properties ?, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21). Additionally, we observe that our methods also apply to modular counting. Among others, we obtain novel intractability results for the problems of counting k-forests and matroid bases modulo a prime p. Furthermore, from an algorithmic point of view, we construct algorithms for the problems of counting k-paths and k-cycles modulo 2 that outperform the best known algorithms for their non-modular counterparts. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime p