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Documents  05C25 | enregistrements trouvés : 3

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Post-edited  Unramified graph covers of finite degree
Li, Winnie (Auteur de la Conférence) | CIRM (Editeur )

Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include
(a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree,
(b) Criteria for Sunada equivalence,
(c) Chebotarev density theorem.
This is a joint work with Hau-Wen Huang.
Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include
(a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree,
(b) Criteria for Sunada equivalence,
(c) Chebotarev density ...

05C25 ; 05C50 ; 11R32 ; 11R44 ; 11R45

The partially disjoint paths problem asks for paths $P_1, \ldots,P_k$ between given pairs of terminals, while certain pairs of paths $P_i$,$P_j$ are required to be disjoint. With the help of combinatorial group theory, we show that, for fixed $k$, this problem can be solved in polynomial time for planar directed graphs. We also discuss related problems. No specific foreknowledge is required.

05C10 ; 05C20 ; 05C25 ; 05C38 ; 68Q25 ; 90C27

Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G, A))$ of the Cayley graph $\Gamma(G, A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam$(G) := max_A$ diam$(\Gamma(G, A))$.
It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$. In 2011, Helfgott and Seress gave a quasipolynomial bound, namely, $O\left (e^{(log n)^{4+\epsilon}}\right )$. We will discuss a recent, much simplified version of the proof.
Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G, A))$ of the Cayley graph $\Gamma(G, A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam$(G) := max_A$ diam$(\Gamma(G, A))$.
It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$. In 2011, ...

20B05 ; 05C25 ; 20B30 ; 20F69 ; 20D60

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