What does the underlined statement mean?

Question

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Since $T_i'$ s are trees it doesn't contain multiple edges. Why it not having the edge from G? What does the statement mean?How do I justify this claim?

Answer 1

I have to assume that your source defines 'graph' in a way that allows two nodes to be directly connected by more than one edge, otherwise the construct '$G'$' wouldn't be a graph [note: other sources would call this a 'multigraph']. Let us proceed as if this is the case:

As $T_1'$, $T_2'$, and $T_3'$ are spanning trees of $G'$, they consist of edges from $G'$, with each pair of nodes connected by at most one edge. Each of these edges may be either an original edge of $G$ or a parallel edge. Let us suppose $T_1'$ (say) contains an original edge of $G$. By disjointness, neither $T_2'$ nor $T_3'$ contains this edge. BUT: It is possible that $T_2'$ (say) contains the matching parallel edge. But that means $T_3'$ can contain neither the original edge nor the parallel edge. Now when we convert back to subgraphs of $G$, we find that $T_1$ and $T_2$ both contain the original edge (in $T_2$'s case because the parallel edge was replaced with the original), but $T_3$ doesn't, so the triple-intersection between that edge's nodes is empty (of edges). This analysis generalizes to all other possible combinations. That is what the underlined sentence means: Since at most two of $T_1'$, $T_2'$, $T_3'$ can contain both an original edge and its parallel, at most two of $T_1$, $T_2$, $T_3$ can contain that original edge.