Discussion
Let \(G\) be an undirected connected graph in which every edge has a positive integer weight. Suppose that every spanning tree in \(G\) has even weight. Which of the following statements is/are TRUE for every such graph \(G\) ?
Answer: D
The true statement is In each cycle \(C\) in \(G\), either all edges in \(C\) have even weight OR all edges in \(C\) have odd weight.
Suppose a cycle had an even edge e and an odd edge f. Take a spanning tree that contains e but not f (start from the cycle minus f and extend it). Removing e splits the tree into two parts, and f, lying on the same cycle, reconnects them, so swapping e for f gives another spanning tree whose weight differs by an odd amount. Both trees cannot be even, a contradiction. So the edges of every cycle share one parity.
The other statements fail on examples. A path of two edges of weight 1 has only one spanning tree, of weight 2, so edges need not be even, and a cycle of odd edges does the same. A triangle of odd edges with a pendant edge of weight 2 has every spanning tree of even weight, yet its edges are neither all even nor all odd.
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