Given a simple graph $G=<V,E>$ with $15 + x$ vertices. This graph has 15 vertices with degree $8$ and $x$ vertices with degree $6$, vertices with the same degree are not connected to each other (i.e: a vertices with degree $6$ can only be connected to vertices with degree $8$ and vice versa). Find $|V|$.
Using the Handshaking lemma I got $2|E|=8*15+6x=>|E|=3x+60$.
From the problem statement we have $x \ge 8$. Then I lost. I have no equations can be used to solve for $x$, and the only inequation I can think of is $|V|-1 <= |E| <= \frac{|V|(|V|-1)}{2}$ but that leads to nothing useful in this case.
How can I find the number of vertices in this case?
This is a bipartite graph. Each edge joins a degree $6$ vertex to a degree $8$ vertex. As there are $15$ degree $8$ vertices, there are $120$ edges. But, can you also express the number of edges in terms of $x$?