Faculty of Mathematics, Physics
and Informatics
Comenius University Bratislava

Seminar of Graph Theory - Kan Hu (27.4.2023)

Thursday 27.4.2023 at 9:50, Lecture room M/213

24. 04. 2023 13.07 hod.
By: Martin Škoviera

Kan Hu (University of Primorska):
Graph embeddings, group factorizations, and skew morphisms

A skew morphism of a finite group $A$ is a permutation $\varphi$ on $A$ fixing the identity element, and for which there exists an integer-valued function $\pi$ on $A$ such that $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$. If $\pi$ is constant on the orbits of $\varphi$, the skew morphism $\varphi$ is called smooth. The theory of skew morphisms is a crucial algebraic tool to investigate symmetric embeddings of graphs into orientable surfaces. In this talk, we will present the most recent progress in the classification problem of skew morphisms of cyclic groups, and show that a cyclic group of order $n$ underlies only smooth skew morphisms if and only if $n=2^en_1$, where $0\leq e\leq 4$ and $n_1$ is a square-free odd number.

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