Let w be the minimum weight among all edge weights in an undirected connected…

Answer: The statement 'e is present in every minimum spanning tree.' is FALSE.

Brief explanation:

• 'There is a minimum spanning tree containing e.' — True. If an MST does not contain e, adding e forms a cycle; swapping e with an equal-weight (or heavier) edge on that cycle yields an MST that contains e.

• 'If e is not in a minimum spanning tree T, then in the cycle formed by adding e to T, all edges have the same weight.' — True. If any edge on that cycle were heavier than w, replacing it by e would produce a lighter spanning tree, contradicting minimality; edges cannot be lighter than w since w is the global minimum.

• 'Every minimum spanning tree has an edge of weight w.' — True. If a MST had no w-edge, adding any w-edge would create a cycle containing an edge heavier than w, which could be replaced to get a lighter tree; contradiction. Thus every MST must include at least one edge of weight w.

• 'e is present in every minimum spanning tree.' — False. A counterexample: a triangle where all three edges have weight w. Any MST uses two of the three edges, so the specific edge e can be omitted in some MSTs. Therefore a particular minimum-weight edge need not belong to every MST.

Conclusion: The false statement is that the specific minimum-weight edge e must be present in every minimum spanning tree.