Consider the following statements:

P: There exists no simple, undirected and connected graph with $80$ vertices and $77$ edges.

Q: All vertices of Euler graph are of even degree.

R: Every simple, undirected, connected and acyclic graph with $50$ vertices has at least two vertices of degree one.

S: There exits a bipartite graph with more than ten vertices which are $2$-colorable.

What is the number of correct statements among the above statements?

Statement P:

For a connected graph with $n$ vertices, minimum number of edges must be:

$n-1$

Here,

$n=80$

Minimum edges required:

$80-1=79$

But given edges are only $77$.

So, such a connected graph cannot exist.

Therefore, P is true.

Statement Q:

In an Euler graph, every vertex must have even degree.

So, Q is true.

Statement R:

A simple, undirected, connected and acyclic graph is a tree.

Every tree with more than one vertex has at least two vertices of degree $1$.

Here, vertices are $50$.

So, R is true.

Statement S:

Every bipartite graph is $2$-colorable.

A bipartite graph with more than ten vertices can exist.

So, S is true.

Therefore, all four statements are correct.

Number of correct statements:

$4$