I think this is true. With consideration to Prim's algorithm, the minimum edge of a vertex is either already in a tree, or will be selected eventually.

I also tried a lot of graph and they all seem correct.

If this statement is False, can someone give me a counter-example?

Thanks.

CodePudding user response：

Your proof is almost there.

Proof 1: Prim's algorithm can start at any vertex, and will immediately select the start vertex's minimum edge, or can select any minimum edge if there is a tie.

Proof 2: In Kruskal's algorithm, one of every vertex's minimum edges will be the first to connect that vertex, and it could be any one of them, if there's a tie, depending on the initial sort.

Proof 2 actually proves a stronger theorem: Every minimum spanning tree will include a minimum edge for every vertex.