Abstracts

 
Estefania González Arroyo
Four dimensional chiral polyhedra with helicoidal faces
Regular polyhedra in $\mathbb{R}^4$ were described by Peter McMullen. The symmetry groups in which there are only isometries that preserve orientation, that are finite subgroups of the special linear group, can be described by functions $f_{a,b}$ with $a$ y $b$ in $\mathcal{Q}$ the group of unitary quaternions. The symmetry group of chiral polyhedra with helicoidal faces is also a "rotational group" therefore can be described with functions $f_{a,b}$. In the talk we describe some of these polyhedra and their symmetry groups.
 
Martin Bachratý
Recursive characterisation of skew morphisms of finite cyclic groups

A skew morphism of a finite group $G$ is an element $\varphi$ of $\textrm{Sym}(G)$ preserving the identity element of $G$ and having the property that for each $a\in G$ there exists a non-negative integer $i_a$ such that $\varphi(ab)=\varphi(a)\varphi^{i_a}(b)$ for all $b\in G$. Skew morphisms play an important role in the study of regular Cayley maps. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available to date. In this talk we show that if a skew morphism $\varphi$ of $\mathbb{Z}_n$ is not an automorphism of $\mathbb{Z}_n$, then it is uniquely determined by an element $h$ of $\mathbb{Z}_n$, a skew morphism $\alpha$ of $\mathbb{Z}_a$ where $a<n$, and a skew morphism $\beta$ of $\mathbb{Z}_b$ where either $b<n$, or $b=n$ and $\textrm{ord}(\beta)<\textrm{ord}(\varphi)$. Conversely, we also list necessary and sufficient conditions for a triple $(h,\alpha,\beta)$ to define a skew morphism of a given cyclic group. In particular, this gives a recursive characterisation of skew morphisms of finite cyclic groups.

The author acknowledges funding from the EU NextGenerationEU through the Recovery and Resilience Plan for Slovakia under the project No. 09I03-03-V04-00272.

 
Robert Daniel Barish
Nondeterministic Constraint Logic (NCL) through the lens of Wang tilings

The Nondeterministic Constraint Logic (NCL) framework (Hearn & Demaine; Theoret. Comput. Sci. 343(1-2); 2005) allows us to express any problem in PSPACE as the task of moving between fully or partially specified orientations of a simple cubic polyhedral graph having integral edge weights (of $1$ or $2$) and bounded bandwidth - i.e., a bounded gap between adjacent vertices in some linear ordering of the graph's vertex set - by way of sequentially reversing the orientations of directed edges under the constraint that all vertices have weighted in-degree at least $2$. Owing to its simple and intuitive nature, the NCL model has since become a cornerstone of PSPACE-completeness reductions for reconfiguration problems in graph theory, computational geometry, and formal models of puzzles and games.

In this talk, we discuss how the graph bandwidth constraint allowed for by the NCL model yields a natural encoding of NCL reachability problems as Wang tiling tasks for finite rectangular or toroidal areas. Concerning a specific application of this correspondence, we also show that the existence of linear time tiling algorithms for sequentially permissive (or ``brick'') Wang tile sets (Derouet-Jourdan, Mizoguchi, & Salvati; Mathematical Progress in Expressive Image Synthesis III. Mathematics for Industry 24; 2016), originally developed for modeling wall patterns, implies novel tractability results for the perfect matching reconfiguration problem (Bonamy et. al.; Proc. 44th MFCS; 2019) of finding restricted walks on the $1$-skeleton of the perfect matching polytope for a graph.

 
Nino Bašić
Nut graphs with a prescribed automorphism group

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry (i.e. are full). We show that every finite group can be represented as the group of automorphisms of infinitely many nut graphs. Moreover, such nut graphs exist even within the class of regular graphs.

This is joint work with Patrick W. Fowler.

 
Jiyong Chen
On the regular embedding of direct product of graphs
A pair of graphs \((\Gamma, \Sigma)\) is stable if \(Aut(\Gamma \times \Sigma) \cong Aut(\Gamma) \times Aut(\Sigma)\). A graph \(\Gamma\) is stable if \((\Gamma, K_2)\) is a stable pair. In 1996, Nedela and Škoviera demonstrated that when \(\Gamma\) is stable, all orientably-regular embeddings of its canonical double cover, \(\Gamma \times K_2\), can be described in terms of regular embeddings of \(\Gamma\). In this talk, we extend this result to the product \(\Gamma \times \Sigma\), where \((\Gamma, \Sigma)\) is a stable pair that satisfies certain conditions.
 
Marston Conder
Recent developments on regular maps and quotients of triangle groups

In this talk I'll describe some recent discoveries (since 2022) about regular maps of small genus (in joint work with Primož Potočnik) and about finite quotients of triangle groups (in joint work with PhD student Darius Young).

As is well known, (ordinary) triangle groups of the form \(\Delta^+(k,l,m) = \langle\, x,y,z\ |\ x^k = y^l = z^m = xyz = 1\,\rangle\) (Equation) play an important role in the study of large automorphism groups of algebraic curves and compact Riemann surfaces, and of regular maps on orientable and non-orientable surfaces.

Much of the early part of the study of these things (dating back over 100 years) considered only small quotients of triangle groups, and subsequent work concentrated on finite simple quotients. But surprisingly, the recent determination of all orientably-regular maps of genus up to 1501 (by MC & PP) has shown that finite simple groups and finite insoluble groups account respectively for less than 0.1% and less than 7% of the associated quotients of ordinary triangle groups, while finite soluble quotients account for over 93%. Some other information is also interesting, such as the increasing proportion of orientably-regular maps of genus $2$ to given $g$ that are chiral. The long-standing question of what happens asymptotically (as $g \to \infty$) remains open.

Very little attention has been paid to soluble quotients, and as one step towards correcting this, a new theorem (by MC & DY, using a 1970 theorem by David Singerman) shows how every non-perfect hyperbolic ordinary triangle group has a smooth finite soluble quotient with derived length $c$ for some $c \le 3$, and has infinitely many such quotients with derived length $d$ for every $d > c$.

Also I will report on some work by Darius in 2024 in which he proved that the natural density (in the positive integers) of the set of orders of finite quotients of a triangle group $\Delta^+(k,l,m)$ is zero for every triple $(k,l,m)$. This work answers a question raised by Tom Tucker (who showed it holds when one of $k,l,m$ is coprime to the other two), and also completes and considerably extends an initial investigation by Larsen (2001), but avoids resorting to Larsen's use of the classification of finite simple groups.

Incidentally, it has also led to unexpected discoveries (by MC & Gabriel Verret & DY) about the densities of finite quotients of free products of two cyclic groups, and hence to the densities of orders of automorphism groups of regular maps of type $\{m,k\}$ for fixed $k$ and variable $q$ (or vice versa), and the densities of orders of finite locally-transitive graphs, and surprisingly, also about possibilities for the density of orders of finite quotients of a given finitely-generated group.

 
Gabe Cunningham
Polytopality criteria for the mix of polytopes and maniplexes

The mix of two maniplexes is the minimal maniplex that covers both. This construction has many important applications, such as finding the smallest regular cover of a maniplex. If one of the maniplexes is an abstract polytope, a natural question to ask is whether the mix is also a polytope. We describe here a general criterion for the polytopality of the mix which generalizes several previously-known polytopality criteria.

This is a joint work with Isabel Hubard.

 
Josse van Dobben de Bruyn
Harmonic homomorphisms, leaking non-abelian flows, and Negami's conjecture

Negami's conjecture states that a graph $X$ admits a planar cover if and only if $X$ is projective. In this talk, I will discuss a relaxation of this problem, where we replace the planar cover of $X$ by the more general notion of a weighted harmonic homomorphism from a planar graph to $X$. Remarkably, the existence of a harmonic homomorphism of certain degree from a planar graph to $X$ is equivalent to the non-existence of certain leaking non-abelian flows in $X$. The latter problem can be formulated and studied as a word problem in a finitely presented group associated to $X$. This gives us a new algebraic tool for tackling Negami's conjecture. Although there are known counterexamples to ``Negami's conjecture with harmonic homomorphisms'', we are nevertheless able to use our techniques to prove new results about the remaining open cases of Negami's conjecture. In particular, we show that $K_{1,2,2,2}$ has no $k$-fold planar cover when $k \not\equiv 0 \pmod 6$.

This talk is based on joint work with David E. Roberson and on related work of Slofstra and Zhang.

 
Zdeněk Dvořák
Crossing number and graph coloring
Famously, every planar graph is $4$-colorable, and every triangle-free planar graph is $3$-colorable. Motivated by these intriguing facts, it is natural to ask what can be said about coloring of graphs that are "nearly planar", in the sense that they can be drawn in the plane with a bounded number of crossings. In this talk, I will survey results and open problems concerning this topic.
 
Mark Ellingham
Parity-based conditions for graph embeddings and their twisted duals
Being orientable, bipartite, or eulerian are all properties of graph embeddings that involve some kind of parity condition. Any cellular graph embedding is related to other graph embeddings by the twisted duality operations of Ellis-Monaghan and Moffatt, which build on Chmutov's partial duality and the sixfold structure of graph embeddings under duality and Petrie duality due to Wilson and independently to Lins. In this talk we discuss interactions between parity-based conditions and duals or twisted duals. For example, there are only certain allowable combinations of orientability and bipartiteness for the six graphs in the Wilson/Lins structure, and we explain how these arise from a `Fano framework', which can also be extended to incorporate eulerian properties. As another example we show that every embedded graph has a bipartite twisted dual, and give characterizations of which twisted duals are bipartite. Fundamental ideas in our proofs involve parity conditions for closed walks in the graph-encoded map or gem and its extension the jewel, and special types of bidirections (orientations of half-edges) in the embedded medial graph. This is joint work with Blake Dunshee.
 
Rhys J. Evans
Orderly generation of highly symmetrical discrete objects

Collections of small mathematical objects are often used in the formation and testing of conjectures, and can also provide counterexamples to open problems. In general, the enumeration of discrete objects is computationally hard. However, additional structure and symmetries can aid in the implementation of common isomorph-free exhaustive generation algorithms. For many highly symmetrical discrete objects, their description as a finite group together with a generating set with certain properties is often useful computationally and theoretically (e.g., regular maps, maniplexes and Cayley graphs).

In this talk, we will see the application of an orderly generation algorithm to the enumeration of minimal generating sets of a given group. Simple group-theoretical observations will be used to improve on the basic algorithm, extending previous enumerations to groups of much larger order. This approach has been used to generate a complete list of minimal Cayley graphs on up to 511 vertices. I will also briefly mention applications to other highly symmetrical discrete objects, and the databases and software we are developing to make the resulting collections of objects available to a wider audience.

 
Lukáš Gáborik
Flow triangulations of cubic graphs
Mattiolo et al., alongside researching complex flows, introduced a representation of the two-dimensional nowhere-zero flow on a cubic graph by a collection of triangles, so-called flow triangulation. In this talk, we examine them further, noticing that sometimes the flow triangulation may induce a tessellation of the plane. Such triangulations contain more geometric context of the flow, which is promising in calculating upper bounds on the flow number exactly. Moreover, we show that their existence depends on whether the graph is toroidal. We also develop an algorithm for finding a flow triangulation without overlaps.
 
Štefánia Glevitzká
Cubic girth-regular graphs of girth six

Recall that given a graph $\Gamma$, the girth of $\Gamma$ is the length of a shortest cycle in $\Gamma$. In graphs of finite girth, each vertex $v$ of $\Gamma$ can be associated with a non-decreasing sequence of integers, one integer for each edge incident with $v$, representing the number of girth cycles containing that edge (possibly $0$). If the sequences associated with the vertices of $\Gamma$ are all identical (and so $\Gamma$ is regular), we say that $\Gamma$ is girth-regular, and the shared sequence is said to be the signature of $\Gamma$. Note that all vertex-transitive graphs are necessarily girth-regular. The concept of girth-regularity was introduced by Potočnik and Vidali in 2019, who also classified cubic girth-regular graphs of girth at most $5$ and cubic vertex-transitive graphs of girth $6$ with respect to their signatures. In our work, we extend the latter to a classification of all cubic girth-regular graphs of girth $6$. We note that for most of attainable signatures, the only girth-regular graphs attaining given signature are the skeletons of maps of type $\{6,3\}$ on the torus or the Klein bottle.

This is a joint work with Robert Jajcay, Maruša Lekse and Primož Potočnik, and it is supported by grants VEGA 1/0437/23 and UK/1398/2025.

 
Štefan Gyürki
Small symmetric directed strongly regular graphs

The notion of a directed strongly regular graph was introduced by Duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case. A directed strongly regular graph (DSRG) with parameters $(n,k,t,\lambda,\mu)$ is a regular directed graph on $n$ vertices with degree $k$, such that every vertex is incident with $t$ undirected edges, and the number of paths of length 2 directed from a vertex $u$ to another vertex $v$ is $\lambda$, if there is an arc from $u$ to $v$, and $\mu$ otherwise. Similarly as in the undirected case, the feasible parameter sets can behave very differently making the efforts for creating catalogues of such (di)graphs difficult.

In the talk we report about a systematic search for symmetric directed strongly regular graphs of small order.

This research was supported from the APVV Research grants 22-0005, 23-0076 and VEGA Research grants 1/0069/23 and 1/0011/25.

 
Kan Hu
A characterization of nilpotent bicyclic groups
A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group theory, with applications extending to symmetric embedding of the complete bipartite graphs, such as regular dessins with underlying graphs $K_{m,n}$. A classical result by Douglas establishes that every bicyclic group is supersolvable. More recently, Fan and Li (2018) proved every finite $(m,n)$-bicyclic group is abelian if and only if $\gcd(m,\phi(n))=\gcd(n,\phi(m))=1$, where $\phi$ is Euler's totient function. In this talk we generalize this result further: we show that every $(m,n)$-bicyclic group is nilpotent if and only if $\gcd(n,\phi(\mathrm{rad}(m)))=\gcd(m,\phi(\mathrm{rad}(n)))=1$, where $\mathrm{rad}(m)$ denotes the radical of $m$ (the product of its distinct prime divisors).
 
Róbert Jajcay
Skew-Morphisms of Regular and Half-Regular Orientable Embeddings of Complete Graphs

Since the classical classification of orientably regular embeddings of complete graphs due to James and Jones preceded the introduction of skew-morphisms, their role in these embeddings has never been fully investigated. In our talk, we plan to revisit these classical results from the point of view of skew-morphisms and to expand this approach to two closely related problems: orientably regular embeddings of complete graphs with multiple edges and half-regular orientable embeddings of complete graphs. The first of these problems has been recently addressed by Gyürki, Pavlíková and Širáň who have been able to classify orientably regular embeddings of complete graphs with multiple edges; yet without the use of skew-morphisms again. We propose to remedy this omission by considering skew-morphisms with extended power functions introduced by Jajcay and Hu. On the other hand, the study of half-regular Cayley maps initiated by Jajcay and Nedela relies on the use of skew-morphisms from its very beginning. Our aim is to apply the skew-morphism based theory to half-regular orientable embeddings of complete graphs with or without multiple edges, and to eventually obtain a classification of orientable half-regular embeddings of complete graphs similar to those of James and Jones and Gyürki, Pavlíková and Širáň.

This is a joint project with Stephen Ikechukwu Ifeanyi.

 
Pavol Jánoš
On $G$-graphs as lifts of dipoles

Graph constructions based on groups have proved invaluable in the search for graphs with given properties, especially in the degree-girth problem, aiming to find the smallest $k$-regular graph of girth $g$. One prominent example is lifting constructions, introduced by Gross and Tucker in 1987, which can be regarded as a generalisation of the well-known Cayley graphs. However, the introduction of $G$-graphs by Bretto and Faisant in 2005, as a new group-based construction that produces highly regular graphs by leveraging group coset structures, has raised a natural question about the relationship between these two approaches. In this talk, we compare the two constructions and derive a sufficient condition under which a $G$-graph can be obtained as a lift of a dipole. Using this result, we further generalise constructions of two families of near-cages of girth $6$ and $8$ to broader families.

This is a joint work with Štefan Gyürki and Jozef Širáň, and it is supported from the APVV Research grants 22-0005, 23-0076 and VEGA Research grants 1/0069/23 and 1/0011/25.

 
Gareth A. Jones
Some regularity properties of maps and hypermaps
I will discuss two recent developments concerning regularity properties of maps and hypermaps. The first is joint work with Martin Mačaj, completing the classifications of orientably or fully regular maps and hypermaps with an automorphism group acting primitively on the vertices. The second is the extension of early work of Wendy Hall on the Macbeath-Hurwitz maps (orientably regular maps of type $\{3,7\}$ obtained from $\mathrm{PSL}_2(q))$ to such maps of type $\{3,n\}$ for all $n>7$; results from number theory are used to determine which of these maps have non-orientable regular quotients of the same type.
 
Ján Karabáš
Classification of group actions on surfaces of genus $\leq 9$

We will discuss the list of representatives of equivalence classes of group actions on surfaces of genera $g$, $2\leq g\leq 9$. I will explain some important examples from the list in details. I will also compare our classification with the known classifications of group actions on surfaces of genera $\leq 5$ and with Breuer's classification of group actions (derived using character theory). Note that any map $M$ of genus $g$ with $\operatorname{Aut}^+M\cong G$ can be viewed as a model of an action of $G$ on a surface $S_g$ with genus $g$. In particular, every action of $G$ on $S_g$ can be obtained by the construction of a Cayley map, and actions with triangular signatures are determined by regular hypermaps. In general, it is difficult to decide whether two Cayley maps determine actions that are topologically equivalent. On the other hand, the topological equivalence of groups with triangular signatures corresponds to the equivalence on orientably regular hyperemaps given by hypermap isomorphisms and dualities. This gives an evidence that the approach taken by Conder and others to classify orientably regular hypermaps is the right one.

This talk is based on a joint work with R. Nedela and M. Skyvová, for details see Journal of Pure and Applied Algebra 228(2024) 107578.

 
Maruša Lekše
Cubic vertex-transitive graphs of girth $7$

Let $\Gamma$ be a graph, and let $v$ be a vertex and $e$ an edge of $\Gamma$. The signature of $e$ is the number of girth cycles that contain it, while the signature of $v$ is the tuple of the signatures of all edges incident to it (ordered by size). We say that $\Gamma$ is girth-regular if every vertex in the graph has the same signature. This concept was introduced by Potočnik and Vidali in 2019 as a generalization of edge-girth-regularity, and they later used it to classify cubic vertex-transitive graphs of girth at most $6$. In this talk, we present a similar classification of cubic vertex-transitive graphs of girth $7$ based on their signatures, along with some corollaries of this classification.

This is joint work with Primož Potočnik and Micael Toledo.

 
Hoi Ping Luk
Classification of Edge-to-edge Monohedral Tilings of the Sphere
Tilings have kindled fascinations across civilisations since antiquity. The studies of spherical tilings can be traced back to Plato. Regarded as spherical analogues of polyhedra, the five Platonic solids give rise to spherical tilings by congruent regular spherical polygons via projection. Beyond them, the quest to classify all the edge-to-edge tilings of the sphere by congruent polygons was pioneered by Duncan Sommerville in 1923. Euler's polyhedral formula asserts that the polygons are triangles, quadrilaterals, or pentagons. The tilings by congruent triangles were eventually classified by Yukako Ueno and Yoshio Agaoka in 2002. Significant progress has since been made in the directions of quadrilaterals and pentagons. The outstanding cases are nevertheless the most challenging and also of the utmost importance as they give rise to the majority of the tilings in question. We have recently solved all the outstanding cases and hence completed the whole classification celebrating the problem's centenary. The talk is based on joint works with Ho Man Cheung and Min Yan.
 
Martin Mačaj
Chirality versus full regularity for orientably regular hypermaps with primitive automorphism groups

Recently G. Jones and M. Mačaj classified the orientably regular hypermaps $\mathcal{H}$ for which the automorphism group $G$ acts primitively and faithfully on the vertices. They showed that they are in one-to-one correspondence with generalised Paley dessins, in which the black vertices are the elements of a finite field $F_q$, and $G$ is a subgroup of the affine group $AG_1(q)$. They also proved necessary and sufficient conditions for these hypermaps to be chiral.

We use these conditions to investigate the balance between reality and chirality for such hypermaps. Since we are dealing with infinite sets of hypermaps, the method of sampling is significant, and various approaches are possible. They all indicate that chirality predominates, though by different amounts. This is a joint work with G. Jones.

 
Antonio Montero
Symmetries of voltage operations in maniplexes and polytopes
A classical result in the theory of maps on surfaces states that the medial of a regular map has two orbits on flags (e.g., it has two types of faces). If the original map is self-dual, however, its medial is regular — it admits symmetry that is somewhat unexpected. This prompts a natural analogue for truncation: what is the symmetry-type of the truncation of a given (say, regular) map? What conditions on the original map guarantee or forbid the emergence of unexpected symmetries for its truncation? Although the answer is known, it is less intuitive than in the medial case. In joint work with Hubard and Mochán, we introduced the notion of voltage operations, which generalise many classical operations on maps and polyhedra to the broader setting of maniplexes and abstract polytopes. In this talk, I will revisit voltage operations and present some results and ideas concerning the central question: when does an operation yield a result with unexpected symmetry?
 
Roman Nedela
Topological equivalence of finite group actions on Riemann surfaces

By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable surface $S_g$ of genus $g\geq 2$. With each such action of a group $G$ on $S_g$, one can associate the fundamental group $\pi(O)$ of the quotient orbifold $O=S_g/G$. It is well-known that $\pi(O)$ is isomorphic to a Fuchsian group $F$ determined completely by orbifold's signature. The Riemann existence theorem reduces the problem of the existence of an action of $G$ on $S_g$ to a group-theoretical problem of deciding whether there is a smooth epimorphism mapping $F$ onto the group $G$. Using computer algebra systems such as Magma or GAP, together with the library of small groups, the generation of all finite group actions on a surface of fixed small genus $g\geq 2$ becomes almost a routine procedure. The difficult part is to determine the classes of these actions with respect to several natural equivalences. One of the most important equivalences is the topological one defined as follows. For an orientable surface $S$ denote by $Hom^+(S)$ the group of orientation-preserving homeomorphisms. Two actions of a group $G$, given by embeddings $\epsilon_i: G \to Hom^+(S_i)$, $i =1, 2$, on possibly different surfaces $S_1$, $S_2$ of the same genus, are topologically equivalent if there is an intertwining orientation preserving homeomorphism $h: S_1\to S_2$ and an automorphism $a\in Aut(G)$ such that $\epsilon_2(g)h =h\epsilon_1(a(g))$, for every $g\in G$. In my talk the following problem is considered.

Problem 1. For a fixed small integer $g>1$, derive the list of actions of finite groups on an orientable surface of genus $g$, distinguished up to topological equivalence.

LLoyd proved that the above problem reduces to a purely group theoretical problem of determining representatives of equivalence classes on the set of smooth epimorphisms from a Fuchsian group $F$ to $G$ defined as follows: two such epimorphisms $\nu_1$, $\nu_2$ are equivalent if and only if there are automorphisms $\alpha\in Aut(F)$ and $a\in G$ such that $\nu_2=a\nu_1\alpha$. To achieve this, one needs to investigate the action of the automorphism group of a Fuchsian group on the set smooth epimorphisms $F\to G$. In order to apply LLoyd's theorem, one needs to derive a finite generating set of $Aut(F)$, where each generator is defined as a transformation of the standard generating set of $F$. Employing known results on generation of $Out(F)$ for surface groups (McCool 1996) and for Fuchsian groups of planar signature (Zieshang 1966), we were able to derive complete lists of finite group actions of genus $g\leq 9$ distinguished up to the topological equivalence.

This talk is based on a joint work with J. Karabáš and M. Skyvová, for details see Journal of Pure and Applied Algebra 228(2024) 107578.

 
Eugenia O'Reilly-Regueiro
Graphs on the flags of designs, and their spectra

In 2012 A. Blokhuis and A.E. Brower defined a graph on the flags of a biplane and proved that the graph corresponding to the unique $(11,5,2)$ is defined by its spectrum. We will look at these graphs for other biplanes, other graphs on flags of designs (symmetric and non-symmetric), some of their properties, and what we can say so far about their spectra.

This is a joint work with Octavio Baltasar Zapata-Fonseca.

 
Claudio Alexandre Guerra Silva Gomes da Piedade
Merging Coset Geometries
Incidence geometries are in the basis of Tits buildings and related structures. Coset incidence systems are incidence structures derived from group cosets, where points, lines, and higher-dimensional elements correspond to cosets of certain subgroups. These capture symmetry and combinatorial properties of groups, particularly in relation to buildings and flag complexes. Given two groups $A$ and $B$, one can define different groups using group operations such as free products (with amalgamation), HNN-extensions, or split extensions. In this talk, given two coset incidence systems of the groups $A$ and $B$, we will describe two distinct ways of building a coset incidence system for its free product (with amalgamation), HNN-extension, and split extension. One of these constructions generalizes the twisting operation on polytopes [1], which is largely inspired on twisted simple groups.

References

  1. McMullen, Peter, and Egon Schulte. Abstract regular polytopes. Vol. 92. Cambridge University Press, 2002.
 
Tomaž Pisanski
On polycirculant drawings of graphs and the House of Graphs
A polycirculant drawing of a graph $G$ is a planar representation of $G$ that remains invariant under a non-trivial rotation, with each vertex orbit having the same size, denoted $m$. A graph $G$ paired with its semi-regular automorphism $a $ is termed a polycirculant structure $(G, a)$. The House of Graphs has recently introduced a new format for the input and output of graphs along with their vertex coordinates. This feature allows computer-generated drawings of graphs with precisely specified vertex coordinates which makes familiar graphs and graph families visually recognisable. In this talk, we focus on drawing polycirculant graphs, particularly those with a Hamiltonian cycle compatible with a given polycirculant structure. The presentation draws on my recent and ongoing work with coauthors, including Leah Berman, Simona Bonvicini, Gábor Gévay, Jan Goedgebeur, Gauvain Devillez and Arjana Žitnik. This research is supported in part by the Slovenian Research Agency through research program P1-0294 and projects J1-4351, J5-4596, and BI-HR/23-24-012.
 
Primož Potočnik
Datasets of regular maps and their skeletons

Datasets containing lists of mathematical objects of interest are a valuable research tool as they can suggest possible theorems and provide counterexamples to existing conjectures. In algebraic graph theory, such datasets have a long history, going back to the 1930s and early versions of the Foster census of cubic symmetric graphs. In the field of highly symmetrical maps, early efforts to compile datasets of various kinds were made by several renowned mathematicians, including Brahana, Coxeter, and most notably Steve Wilson. In the modern computer era, this work was taken significantly further by Marston Conder, whose lists of all regular and chiral maps up to a given genus have long served as an invaluable resource for the community.

In my talk, I will present some recent joint work with Marston Conder that takes these censuses to a new level. Specifically, I will describe a census of all regular (both orientable and non-orientable) and chiral maps of genus up to 1.501 (1.502 in the non-orientable case), or with no more than 3,000 edges (6,000 edges in the orientable case). I will explain the techniques that enabled us to extend the existing censuses and illustrate some recent applications.

 
Jozef Rajník
A geometric approach to lower bounds on the complex flow number

A complex nowhere-zero $r$-flow on a graph $G$ is a flow using complex numbers whose Euclidean norm lies in the interval $[1,r-1]$. The complex flow number of a bridgeless graph $G$, denoted by $\phi_{\mathbb{C}}(G)$, is the minimum of the real numbers $r$ such that $G$ admits a complex nowhere-zero $r$-flow. Complex (and, more generally, multidimensional) flows comprise an interesting generalisation of the classical integer and circular flows. However, the exact computation of $\phi_{\mathbb{C}}$ seems to be a difficult task even for very small and symmetric graphs, including the Petersen graph.

In this talk, we present two lower bounds on the complex flow number of cubic graphs: one general depending on the odd girth, and another for the infinite family of Isaacs' flower snarks. The proofs of these lower bounds are based on a convenient geometric representation of a complex flow by a set of suitable triangles in the plane. This representation also uncovers interesting connections to duals of the considered graphs embedded to other surfaces.

This is joint work with Davide Mattiolo, Giuseppe Mazzuoccolo and Gloria Tabarelli. Funded by the EU NextGenerationEU through the Recovery and Resilience Plan for Slovakia under the project No. 09I03-03-V05-00012.

 
Haofang Sun
Tiling of the sphere by curvilinear quadrilaterals
Edge-to-edge tilings of the sphere by congruent quadrilaterals, where the quadrilaterals have straight edges, have been completely classified. We study the same problem, for the so-called curvilinear quadrilaterals, where the edges are not necessarily straight. Similar to the case of straight edges, there are four types of curvilinear quadrilaterals ($A^2BC$, $A^3B$, $A^2B^2$, $A^4$) that are suitable for tiling. In this talk, we classify tilings of the sphere by congruent curvilinear quadrilaterals of types $A^3B$ and $A^2BC$. For type $A^3B$, we use the "straightening" method to show the non-existence of some tilings that are combinatorially possible.
 
Micael Toledo
Highly symmetric unstable maniplexes
A maniplex of rank $n$ is an $n$-valent properly edge-coloured graph that generalises, simultaneously, maps on surfaces and abstract polytopes.he T problem of stability in maniplexes is a natural variant of the problem of stability in graphs in general. A maniplex is stable if every automorphism of its canonical double cover is a lift of some automorphism of the original maniplex. Due to their very rich structure, regular (maximally symmetric) maniplexes are always stable. It is thus natural to ask what is the maximum possible degree of symmetry that a maniplex that is not stable can admit.
 
Thomas Tucker
Regular Map Type
TBD
 
Stephen E. Wilson
Some Constructions of Symmetric Hypergraphs
The talk will present four constructions of symmetric hypergraphs and indicate which of them might underlie a rotary hypermap.
 
Min Yan
Vertex type and angle congruent tiling

In the classification of tilings of a surface by certain prototiles, the first step is usually the determination of all the possible combinations of the angles at vertices. For example, this is the first step in Rao’s classification of convex pentagons that can tile the plane. We discuss how this can be done for tilings of the sphere by angle congruent pentagons, and compare our approach to Rao's. Then we apply the results to get the classifications of some families of tilings of the sphere by angle congruent pentagons.

This is a joint work with Robert Barish and Hoi Ping Luk, and is supported by NSFC/RGC Joint Research Scheme N_HKUST607/23.

 
Yifan Zhang
Simplicial approximation of PL eliminations of intersections
Many constructions in topological combinatorics rely on piecewise-linear (PL) maps, but for computational or combinatorial applications, it is often desirable to realize these maps simplicially. In this talk, we explore the process of replacing PL constructions on handle cancellation with simplicial ones through iterated subdivisions while preserving the intersection behaviour. As a guiding application, we consider the counterexample to the topological Tverberg conjecture proposed by Mabillard and Wagner in sufficiently large dimensions. We outline how the key local operations such as higher-mulitplicity Whitney tricks can be made simplicial and discuss the complexity of this process.