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G is a graph on n vertices and 2n-2 edges. The edges of G can be partitioned into two edge-disjoint spanning trees. Which of the following is NOT true for G?

1. For every subset of k vertices, the induced subgraph has at most 2k-2 edges
2. The minimum cut in G has at least two edges
3. There are two edge-disjoint paths between every pair of vertices
4. There are two vertex-disjoint paths between every pair of vertices

Correct answer: There are two vertex-disjoint paths between every pair of vertices

Solution

The statement about having two vertex-disjoint paths between every pair of vertices is not necessarily true because while G has two edge-disjoint spanning trees, this does not guarantee that there are vertex-disjoint paths, as the paths may share vertices.

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