The achromatic number of Knesser graphs and their relationship with Steiner triple systems

Gabriela Araujo Pardo, Mathematics Institute, National University of Mexico

In this talk we give the notion of complete colorings in graphs, achromatic number, Knesser graphs and Steiner triple systems . Also, we explain how the Steiner triple systems solve the problem about the existence of complete colorations on Knesser graphs that attain the upper bound of the achromatic number, where the achromatic number of a graph G is the maximum integer value for the number of chromatic classes in a complete and proper coloring of G.

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Open problems on k-orbit polytopes (slides)

Gabe Cunningham, University of Massachusetts Boston

A k-orbit abstract polytope is one where the action of the symmetry group on the flags has k orbits. When k = 1, these are the regular polytopes, which have been extensively studied. There is also a growing body of research for the case k = 2. However, very little is known about k-orbit polytopes with k ≥ 3. In this talk, I will discuss some of the difficulties that arise when trying to work with k ≥ 3. I will also present a broad range of fundamental questions about k-orbit polytopes, encompassing combinatorial, geometric, and algebraic aspects.

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Intersection properties for k-orbit polytopes (slides)

Elías Mochán, Institutode Matemáticas - UNAM

We will define the flag graph of a polytope and describe how to recover the polytope from its flag graph. We will also define the symmetry type graph of a polytope with respect to a group of automorphisms and use voltage assignments to recover the flag graph from the symmetry type graph and the corresponding group. We will use this construction to characterize the groups that act on some polytope with a given symmetry type graph. This groups are characterized by certain intersection properties on some distinguished subgroups and some of their cosets.

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Connection groups of pyramids over regular 3-tori (slides)

Gordon Ian Williams Williams, University of Alaska Fairbanks

We will discuss the results and techniques of a recent investigation into the connection (or monodromy) groups of the pyramids over the regular 3-tori. Of particular note is the discovery of an infinite family of of 4-polytopes whose connection groups are not string C-groups, and the development of a method for representing the connection group of a polytope as a subgroup of a wreath product of its automorphism group with the symmetric group.

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Local Theory in Tilings, Delone Sets, and Polytopes (slides)

Egon Schulte, Northeastern University

Local detection of a global property in a geometric or combinatorial structure is usually a challenging problem. The Local Theorem for Tilings says that a tiling of Euclidean d-space is tile-transitive (isohedral) if and only if the large enough neighborhoods of tiles (coronas) satisfy certain conditions. This is closely related to the Local Theorem for Delone Sets, which locally characterizes the uniformly discrete point sets in d-space which are orbits under a crystallographic group. Both results are of great interest in crystallography. We discuss old and new results from the local theory of tilings and Delone sets, and point to interesting open problems for polytopes.

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Jumping in half-arc-transitive graphs

Alejandra Ramos Rivera, University of Primorska, IAM & FAMNIT

In this talk we focus on tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, that is a subgroup acting transitively on its vertices and its edges but not on its arcs. One of the most fruitful approaches for the study of structural properties of such graphs is the well known paradigm of alternating cycles and their intersections which was introduced by Marušič 20 years ago.

We introduce a new parameter for such graphs, the alternating jump parameter, giving a further insight into their structure. The obtained results are used to establish a link between two frameworks for a possible classification of all tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, the one proposed by Marušič and Praeger in 1999, and the much more recent one proposed by Al-bar, Al-kenai, Muthana, Praeger and Spiga which is based on the normal quotients method.

We also present results on the graph of alternating cycles of a tetravalent graph admitting a half-arc-transitive subgroup of automorphisms. A considerable step towards the complete answer to the question of whether the attachment number necessarily divides the radius in tetravalent half-arc-transitive graphs is made.

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Biquasiprimitive Oriented Graphs of Valency Four

Nemanja Poznanovic, University of Melbourne

(Joint work with Cheryl Praeger)

In this talk I will describe a recently initiated research programme aiming to use a normal quotient reduction to analyse finite connected, oriented graphs of valency 4, admitting a vertex- and edge-transitive group of automorphisms which preserves the edge orientation. In the first article on this topic [1], a subfamily of these graphs was identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one ‘basic’ member. These basic members can be further divided into three types: quasiprimitive, biquasiprimitive and cycle type. The first and third of these types have been analysed in some detail in the papers [1, 2, 3]. Recently, we have begun an analysis of the basic graphs of biquasiprimitive type. I will describe this approach and mention some results.

[1]. Al-bar, J. A., Al-kenani, A. N., Muthana, N. M., Praeger, C. E., and Spiga, P. (2016). Finite edge-transitive oriented graphs of valency four: a global approach. Electr. J. Combin. 23 (1), #P1.10. arXiv: 1507.02170.

[2]. Al-Bar, J. A., Al-Kenani, A. N., Muthana, N. M., and Praeger, C. E. (2017). Finite edgetransitive oriented graphs of valency four with cyclic normal quotients. Journal of Algebraic Combinatorics, 46(1), 109–133.

[3]. Al-bar, J. A., Al-kenani, A. N., Muthana, N. M., and Praeger, C. E. (2017). A normal quotient analysis for some families of oriented four-valent graphs. Ars Mathematica Contemporanea, 12(2), 361–381.

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Chiral polytopes of arbitrarily large rank

Marston Conder, University of Auckland

Regular abstract polytopes are well understood and relatively easy to construct. In contrast, chiral polytopes are much more challenging, and indeed less than 15 years ago, no finite chiral polytopes of rank d ≥ 5 were known. That changed in the mid/late 2000s when some examples of ranks 5 to 8 were found, and then Daniel Pellicer (2010) proved the existence of finite chiral polytopes of arbitrarily large rank d. Daniel's examples were enormous, however, and it has been a challenge to find some smaller concrete examples. Gabe Cunningham recently proved that for d ≥ 8, every chiral d-polytope has at least 48(d-2)(d-2)! flags, and it is no longer surprising that 'small' examples of large rank do not exist. Nevertheless, examples are now plentiful. In this talk I will describe a new construction (developed in joint work with Isabel Hubard, Eugenia O'Reilly-Regueiro and Daniel Pellicer) that can be used to show that for every d ≥ 5, there exists a chiral d-polytope with automorphism group isomorphic to the alternating group An and a chiral d-polytope with automorphism group isomorphic to the symmetric group Sn, for all but finitely many n.

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Chiral extensions of toroids

Antonio Montero, Centro de Ciencias Matemáticas - UNAM

An abstract politope K is an extension of an abstract polytope P if all the facets of K are isomorphic to P. Chiral polytopes are those that have all possible rotations but lack of reflectional symmetry. In the talk we will explore the problem of building chiral extensions of polytopes. We will also show a technique to build chiral extensions of regular toroids.

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Free extensions of two-orbit polytopes

José Collins, UNAM

An extension of a n-polytope P is a (n+1)-polytope with every facet isomorphic to P. P is a two-orbit polytope in class 2I if its automorphism group induces exactly two orbits on its set of flags such that a flag and its i-adjacent flag are in the same orbit if and only if i ∈ I. In this talk we show that, for some classes I ⊊ {0,... ,n-1}, every n-polytope in class 2I has an extension in class 2I such that every other extension in the same class can be obtained from it by means of suitable identifications.

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CPR Graphs of Toroidal Regular Maps of Type {4, 4} (slides)

Maria Elisa Fernandes Fernandes, Universidade de Aveiro

CPR graphs, that are faithful permutation representations of string C-groups, turned out to be a powerful tool in classification of abstract regular polytopes. We will determine all possible degrees of a CPR graph of the toroidal maps of type {4, 4}.

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Hypertopes with tetrahedral diagram (slides)

Domenico Catalano, University of Aveiro

We give an infinite family of hypertopes of rank four having tetrahedral diagram. Their group of rotational symmetries is isomorphic to PSL(2,q). It turns out some elements of this family are regular hypertopes and some are chiral. Moreover, we show that the chiral ones have both improper and proper correlations simultaneously.

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On The Existence of Finite Groups in Which Certain Distinguished Generators and Products of Pairs of Generators Have Prescribed Order

James Fraser, Open University

In this talk we shall provide a method for constructing finite groups in which certain distinguished generators and products of pairs of generators have prescribed orders. To provide the motivation for this we begin by introducing two problems in regular maps which we address with this method. In one case providing a resolution, and in the other partial progress. Both problems ask for which sets of parameters it is possible to construct various regular maps, these parameters being naturally described as orders of certain generators. We proceed by assuming that we can find a finite group with a particular set of parameters as the subgroup of some finite linear fractional group. This in turn gives rise to polynomial equations to which we need to find roots of given orders in finite fields. By providing a resolution to this question we will be able to demonstrate the existence of the desired finite groups.

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Representing automorphism of graphs

Klara Stokes, University of Skövde

There is a natural representation of the automorphism group of a graph in its homology. In this talk I will describe this representation and show how to use it to deduce interesting properties of the quotient graph. I also consider analogies of Weierstrass gaps of graphs and discuss similarities and differences compared to the case of algebraic curves. This is joint work with Milagros Izquierdo.

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Complete regular dessins and skew-morphisms of cyclic groups (slides)

Martin Skoviera, Comenius University, Bratislava, Slovakia

A dessin is a 2-cell embedding of a connected bipartite graph into a closed oriented surface, endowed with a fixed vertex 2-colouring. A dessin is regular if its automorphism group is regular on the edges. Dessins were introduced by Grothendieck over thirty years ago as a combinatorial counterpart of algebraic curves. In this talk we deal with regular dessins whose underlying graph is a complete bipartite graph K{m,n}, called (m,n)-complete dessins. We discuss a rather surprising correspondence between (m,n)-complete regular dessins and skew-morphisms of cyclic groups. We show that every (m,n)-complete regular dessin D gives rise to a closely related "reciprocal" pair of skew-morphisms ZnZn and ZmZm. Conversely, D can be easily reconstructed from the reciprocal pair up to isomorphism. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and reciprocal pairs of skew-morphisms of cyclic groups are all in a one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers m and n for which there exists, up to interchange of colours, exactly one (m,n)-complete regular dessin. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order m and n is abelian, which eventually comes down to the condition gcd(m, φ(n)) = gcd(φ(m), n) = 1.

This is a joint work with Y.-Q. Feng, K. Hu, N.-E. Wang, and R. Nedela.

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Regular self-dual and self-Petrie dual map of arbitrary degree (slides)

Olivia Jeans, Open University UK

(J. Siran, J. Fraser and O. Jeans)

In "Trinity symmetry and kaleidoscopic regular maps" (2014), Archdeacon, Conder and Siran proved that for every even k there exists a regular self-dual and self-Petrie-dual map of degree k. By a combination of techniques from algebraic number theory and covering spaces we extend this result to include all odd k > 4.

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Archimedean Toroids and their Almost Regular Covers (poster)

Mark Mixer, Wentworth Institute of Technology

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