Raziskovalni matematični seminar - Arhiv
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According to a nonstandard definition introduced in 2021, a distance magic labelling ℓ of a regular graph of order n is a bijection from its vertex set to the set of integers of the arithmetic progression from 1-n to n-1 with common difference 2, such that the sum of the labels of the neighbors of each vertex is zero. Such a labeling is called self-reverse if, for any pair of vertices u and v, u is adjacent to v if and only if the vertices with labels -ℓ(u) and -ℓ(v) are adjacent.
In this talk, we present the motivation for studying self-reverse distance magic labelings. We focus on self-reverse distance magic labelings in the case of tetravalent graphs. First, we examine some tetravalent distance magic graphs of small order and determine which of them admit a self-reverse distance magic labeling. We then provide several examples and a complete classification of all orders for which a tetravalent graph admitting such a labeling exists. The classification is obtained via a novel construction that produces a (tetravalent) distance magic graph from two given (tetravalent) distance magic graphs. We also discuss the existence of graphs admitting a self-reverse distance magic labeling among some well-known families of tetravalent graphs and conclude with possible directions for future research.
This is joint work with Petr Kovář and Primož Šparl.
Recently, we have investigated a new environment for the conjecture - that of stacks - the geometric objects generalizing varieties. This perspective has led to unexpected applications beyond its original scope, offering insights into phenomena related to the occurrence of Galois extensions—a topic that, at first glance, seems unrelated to counting solutions of polynomial equations.
The starting point is the use of auxiliary graphs, the so called hexagon graphs, associated with Cayley maps of $(2,s,3)$-groups. Large induced trees in these auxiliary graphs provide a route to Hamilton cycles/paths in the original Cayley graphs. I will then explain why quartic one-regular graphs are central in a possible generalization of this approach to $(2,s,4)$-groups.
In the final part, I will discuss the Feng--Xu normality theorem for quartic Cayley graphs on regular $p$-groups and propose a CFSG-free proof in the special case where one generator has order $p$.
This leads naturally to broader questions about direct proofs in algebraic combinatorics, including primitive groups of degree $2p$ and the Polycirculant conjecture.
Juvan, Malnič and Mohar [J. Combin. Theory Ser. B 68 (1996), 7-22] proved that for every surface S and every integer k, there exists N(S,k) such that any set of simple closed curves on the surface S, where any two are non-homotopic and intersect at most k times, has a maximum size at most N(S,k). In the case when S is the torus T^2, the problem has an interesting connection to number theory as the Riemann hypothesis yields that N(T^2,k)<=k+O(k^0.5*log k); Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] have recently proven that this bound holds directly. We determine N(T^2,k) exactly for every k. In particular, we show that N(T^2,k)<=k+6 for all k and N(T^2,k)<=k+4 when k is sufficiently large.
The talk is based on joint work with Igor Balla, Marek Filakovský, Bartłomiej Kielak and Niklas Schlomberg.
We investigate finite semigroups equipped with a preorder relation that is compatible with the semigroup operation and we develop an enumeration methodology based on automorphism group actions. For a fixed finite semigroup S, the automorphism group Aut(S) acts naturally on the set Pre(S) of compatible preorders via pushforward; the orbits of this action correspond exactly to isomorphism classes of preordered semigroups. Using known classifications of semigroups of small order, together with exhaustive generation of compatible preorders, our method yields a complete enumeration of preordered semigroups up to order five. Finally, this framework opens a computational window toward the study of more complex algebraic systems, particularly EL-hyperstructures arising in hypercompositional algebra, where preordered semigroups serve as a fundamental tool for their construction.
ZOOM link: https://upr-si.zoom.us/j/65127426860?pwd=kYjS6yFt6yeUcYAGGPcnJShbb5UI7r.1
