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Any hint please if I should prove or search for counterexample, at least to know which direction to go...

Consider the following 2 graphs:

G1 (V, E1)
G2 (V, E2)

and a weight function for the vertices w: (E1 \untion E2) -> R

and 2 minimum spanning trees T1, T2 of G1 and G1.

We define a new graph: G(V, E1 \untion E2)

Is it true to say that there is always a minimum spanning tree T in the new graph G (using same w function) such that all edges of T are from T1 or T2?

The claim is wrong in case for every T it has an edge e which isn't included in T1 or T2, Now in G1 we can remove this edge (e) and the problem is solved for T1. But I can't remove edge e from G2 too, which means I must take it and the claim is right, am I wrong?

Note: no need for detailed answer, only looking to know what is correct and giving it myself a deep prove and thinking.

john
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    Hint: compare what Kruskal's algorithm would do on (V, E1 U E2) versus (V, T1 U T2). – David Eisenstat May 07 '21 at 12:16
  • @DavidEisenstat can you kindly read the updated question, on why I think it's true. plus you just wrote (V, T1 U T2) T1 and T2 are not edges. – john May 07 '21 at 12:18
  • I don't follow your proof. "The claim is wrong in case T has an edge e which isn't included in T1 or T2." No, the claim is wrong if for *all* MSTs T, there exists such an edge. The rest I don't understand. – David Eisenstat May 07 '21 at 12:26
  • @DavidEisenstat you are right, just can you guide me which direction to go? (if it's true to not) so I would spend time on proving or finding contradiction example (probably I think it's true) – john May 07 '21 at 12:36
  • Why don't you work some small examples by hand? I think you'll either find a counterexample or get a good feel for how the proof should go. – David Eisenstat May 07 '21 at 12:41
  • I tried many and non worked as counterexample, so probably it's true but I don't "feel" that way eaither – john May 07 '21 at 13:25
  • @DavidEisenstat any hint please? – john May 07 '21 at 14:19
  • Why would Kruskal's algorithm choose an edge in (E1 U E2) \ (T1 U T2)? – David Eisenstat May 07 '21 at 14:26

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