One or both of the edges with the third and the fourth smallest weights are part of any MST simple undirected weighted graph (distinct edge weights)

Question

In a simple undirected weighted graph $G$, all of whose edge weights are distinct. Is the following statement True or false?

One or both of the edges with the third smallest and the fourth smallest weights are part of any minimum spanning tree of $G$

---

Let me explain the problem or issue which I am facing. It is quite easy to prove the statement when the number of edges in the graph is at least four. The place where I am having the issue is considering the corner cases.

Usually the text books do not deal with these kind of corner cases, so as such I am not quite acquainted with the usual convention which is followed.

Does having a connected graph with only $2$ edges, make this statement false? I am not actually getting it. If the graph itself has only two edges, then no point of having the third or fourth minimum edge weight in the MST, as they do not exist.

It is like saying:

One or both of Mr. Robert's daughters go to school.

Now if Mr Robert has no girl child in the first place, then will this statement be true or false.

I felt like interpreting the statement in the actual problem as:

the third smallest edge (if it exists) and the fourth smallest edge (if it exists)...

But since it is a mathematics question, I felt like clarifying once, because what I might assume myself might be contrary to what is followed in usual literature...

Can anyone please help me out as to what happens in the corner case? With logical reasoning (first order logic?) [ In statements involving universal quantifiers, the statement becomes vacuously true if the universe of discourse is empty. While for existential quantifiers, the statement becomes false. But I am unable to figure out the given statement using this manner of thinking]

Answer 1

There is not, and should never be, a convention in the literature about what this means. The statement is simply poorly written, because it does not address the case where the third and fourth edges don't exist.

• If you are writing this statement yourself, don't rely on it being either false or vacuously true in corner cases; call them out specifically. For example, add "Assume $G$ has at least four edges."
• If you are forced to declare the statement "true" or "false" on an exam or similar, make a clarifying comment if you are allowed to, and otherwise do your best to read the examiner's mind.
• If you are encountering the statement while learning on your own, then - well, you've already figured out everything there is to figure out about the statement, so you've done all the math there is to do about it. Quibbling over whether that adds up to the statement being true or false is pointless.