let start from a fixed vertex,name it 1 and then name the next vertex 2 (let clockwise)and the next after that 3 like that way name the vertices as 1,2,........2n.
now start assigning digits to vertices beginning from 1,
assign 1111111...1 (n digits) to the vertex 1.
assign 2111111...1 to the vertex 2.
assign 2211111...1 to vertex 3
go on that way to see
22222....2 is assigned to vertex n+1.
now start doing the opposite from the next vertex
122222...2 to vertex n+2.
11222.....2 to vertex n+3.
go on that way
to see
11111.....12 is assigned to the vertex 2n.
see the numbering for vertices 2,3,....n+1 start with 2.
and for the vertices n+2,....2n,1 start with 1.so they are distinct.
also within each group all numbers are distinct.
so all the 2n numbers are distinct.
rest of the properties follows by the method of construction.