Finite skew braces with isomorphic non-abelian characteristically simple additive and circle groups

A skew brace is a triplet ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , where ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are groups such that the brace relation x ∘ ( y ⋅ z ) = ( x ∘ y ) ⋅ x - 1 ⋅ ( x ∘ z ) x\circ(y\cdot z)=(x\circ y)\cdot x^{-1}\cdot(x\circ z) holds for all x , y , z ∈ A x,y,z\in A . In this paper, we study the number of finite skew braces ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , up to isomorphism, such that ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are both isomorphic to T n T^{n} with 𝑇 non-abelian simple and n ∈ N n\in\mathbb{N} . We prove that it is equal to the number of unlabeled directed graphs on n + 1 n+1 vertices, with one distinguished vertex, and whose underlying undirected graph is a tree. In particular, it depends only on 𝑛 and is independent of 𝑇.

Slit-Slide-Sew Bijections for Bipartite and Quasibipartite Plane Maps

We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different faces), we build a new plane map with a distinguished vertex and two distinguished half-edges directed toward the vertex. The faces of the new map have the same degree as those of the original map, except at the locations of the distinguished corners, where each receives an extra degree: this is the location of the distinguished half-edges. This bijection provides a sampling algorithm for uniform maps with prescribed face degrees and allows to recover Tutte's famous counting formula for bipartite and quasibipartite plane maps. In addition, we explain how to decompose the previous bijection into two more elementary ones, which each transfer a degree from one face of the map to another face. In particular, these transfer bijections are simpler to manipulate than the previous one and this point of view simplifies the proofs.

A Canonical Ramsey Theorem for Exactly m-Coloured Complete Subgraphs

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ $\mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that X\{v} is 1-coloured and each edge between v and X\{v} has a distinct colour (all different to the colour used on X\{v}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

The Lollipop Graph is Determined by its Spectrum

An even (resp. odd) lollipop is the coalescence of a cycle of even (resp. odd) length and a path with pendant vertex as distinguished vertex. It is known that the odd lollipop is determined by its spectrum and the question is asked by W. Haemers, X. Liu and Y. Zhang for the even lollipop. A private communication of Behruz Tayfeh-Rezaie pointed out that an even lollipop with a cycle of length at least $6$ is determined by its spectrum but the result for lollipops with a cycle of length $4$ is still unknown. We give an unified proof for lollipops with a cycle of length not equal to $4$, generalize it for lollipops with a cycle of length $4$ and therefore answer the question. Our proof is essentially based on a method of counting closed walks.

Attaching graphs to pseudo-similar vertices

Vertices u and v of a graph G are pseudo-similar if G – u ≅ G – v, but no automorphisms of G maps u to v. Let H be a graph with a distinguished vertex a. Denote by G(u. H) and G(v. H) the graphs obtained from G and H by identifying vertex a of H with pseudo-similar vertices u and v, respectively, of G. Is it possible for G(u.H) and G(v.H) to be isomorphic graphs? We answer this question in the affirmative by constructing graphs G for which G(u. H)≅ G(v. H).