Speaker: Pro. Sanming  Zhou (The University of Melbourne)

Title1:How  to write mathematics

Time: Tue, Oct.16, 2018 , AM:10:00-11:00

Title1:Perfect  codes in Cayley graphs (I)，(II)，(III)

Time: Wed. Oct.17, 2018 , PM:7:00-8:30

           Thur. Oct.18, 2018 , PM:7:00-8:30

            Fri. Oct.19, 2018 , AM:8:00-9:30  

Location: Room 4318, Building No.4, Wushan Campus

Abstract：  

Let  $G = (V, E)$ be a graph and $t$ a positive integer. A perfect $t$-code in $G$ is  a subset $C$ of $V$ such that every vertex of $G$ is at distance no more than  $t$ to exactly one vertex in $C$. Perfect $t$-codes in the Hamming graph $H(n,  q)$ are precisely $q$-ary perfect $t$-codes of length $n$ in the classical  setting, and those in the Cartesian product $C_q \Box \cdots \Box C_q$ of cycle  $C_q$ with itself $n$ times are precisely $q$-ary perfect $t$-codes of length  $n$ under the Lee metric. A perfect 1-code in a graph is also called an  efficient dominating set or independent perfect dominating set of the  graph.

Since  both $H(n, q)$ and $C_q \Box \cdots \Box C_q$ are Cayley graphs, perfect codes  in Cayley graphs can be considered as generalizations of perfect codes under the  Hamming or Lee metric. Perfect 1-codes in Cayley graphs are also closely related  to tilings of the underlying groups. In these talks I will discuss perfect codes  in Cayley graphs, with an emphasis on perfect 1-codes.