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Sesión especial del Seminario Preguntón

Ponente: R. Jajcay, T. Jajcayova, L. Montejano, M. Skoviera
Institución: Comenius University, Bratislava / UNAM Juriquilla
Cuándo 19/11/2014
de 11:30 a 19:00
Dónde CINNMA (Casa Amarilla), Juriquilla, Querétaro
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11:30 Robert Jajcay
Comenius University, Bratislava, Eslovaquia

Cayley cages

The famous cage problem is the problem of finding the smallest graphs of given degree and girth. Although the
problem arose already in the 1960’s, significant progress has only been made with the introduction of algebraic,
geometric and topological methods and with the advancement of computers within the last decade. In our talk, we
focus on a specific side of the problem, namely on the relation of the original problem to problems of graph
symmetries. Although the role of graph symmetries in extremal graph theory is not yet completely understood,
many of the current record holder cages have been constructed using highly symmetric structures with large
automorphism groups. We will provide a quick overview of such construction and will focus on examples of
constructions stemming from Cayley graphs and related structures. 


12:30 Tatiana Jajcayova
Comenius University, Bratislava, Eslovaquia

Generalized palindromic closures

Palindromic closure of a finite word w is the shortest palindrome having w as a prefix. For ins- tance, Tribonacci word 
u = 0102010010201010200..., that is defined as a fixed point of the substitution h(0) = 01, h(1) = 02, h(2) = 0, can be 
constructed using directive periodic sequence ∆ = (012)^ω by repeated applications of the operation palindromic 
closure and successive additions of letters from ∆. If we replace reflection by other involutory antimorphism θ, we can 
talk about θ-palindromes and θ-palindromic closures. 

In our presentation, we first summarize known results for infinite words that are created by palindromic closures using 
one fixed antimorphism. Then we look at the situation where words are created using a pair of sequences: a sequence 
of letters ∆ = (δ_n) and a sequence of antimorphisms Θ = (θ_n). We study infinite words that are results of θ-palindro-
mic closures, where for θ we successively take elements of Θ. We show that the Thue-Morse word, as well as its ge-
neralization for multi letter alphabet, can be obtained by the above construction. Similar result for other classes of 
words were recently obtained by Blondin Massé et al. 

The language of all words that are created by generalized palindromic closures is invariant with respect to group G 
generated by antimorphisms that are appearing in Θ infinitely many times. Therefore, among these words we can find 
new examples of G-rich words. 

This is a joint work with Edita Pelantová and Štepán Starosta. 


16:00 Luis Montejano
UNAM Juriquilla

Kneser transversals

What is the maximum positive integer n such that every set of n points in R^d has the property that the convex hulls

of all k-set have a transversal d-lambda-plane? In this paper, we investigate this and closely related questions. We 

define a special Kneser hypergraph by using some topological results and the well-known lambda Helly property. We 

relate our question with the chromatic number of the Kneser hypergraph and we establish a connection lambda =1 

with the so called Kneser. This problem is connected with the Gale embeddings, the discrete version of Rado’s 

Problem, and with cyclic polytopes.


17:00 Martin Skoviera
Comenius University, Bratislava, Eslovaquia

Snarks - recent development

Snarks are nontrivial cubic graphs whose edges cannot be properly coloured with three colours. Since their first
occurrence in the 19th century, these graphs have been an object of continuous interest especially because of their
relationship to various important problems in graph theory, such as the four colour problem, Tutteís 5-flow conjecture,
the cycle double cover conjecture, and Fulkersonís conjecture. In spite of great research efforts, their structure and
properties remain largely unknown. Our talk will survey recent development in this area by discussing various
properties of snarks, in particular those related to their composite structure, perfect matchings, flows, edge-
colourings, symmetries, embeddings, and others.


18:00 Brindis


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