for specific random numbers n,m,k obtained an undirected graph
$G_{n,m,k} ,V_{n,m}, E_{n,m,k}$ is defined as follow
$V_{n,m}=\{A|A $ is a subset of ${1,2,....n}$ and $|A|=m \}$,
$E_{n,m,k}=\{\{A,B\}| | A \cap B|=k\}$
where |A| denotes number of elements in A
I need to find number of vertices and sides of $G_{6,3,2}$ and to generalize what condition of n in order $G_{n,3,1} (n \ge4)$ have euler vertices
Here we know that
$n=6,|A|=m=3, |A \cap B|=k=2$
To find vertices from set which each subset have 3 elements we can choose $6 C3= 20$ subset that have 3 element
but to find edges without drawing, suppose the sets is $\{a,b,c\} $ & $\{b,c,e\}$ since n=6, there are $6C1$ pairs?
how can i find the total pair of sets from n integer when we know $|A \cap B|=2$ ?
$6 C3 * 6C1= 20 * 6 =120 $ is the vertex degree,
since we know that vertex degree=2* edges, total edges= 60?
$G_{n,3,1} (n \ge4)$ have euler vertices when degree of all vertices is even
$n=n,|A|=m=3, |A \cap B|=k=1$
$G_{n,3,1}=\frac{nC3 *nC2}{2} $ is integer is sufficient condition?