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I came across this exercise from section 1.3 in Hatcher's "Algebric topology".

Construct finite graphs $X_1$ and $X_2$ having a common finite-sheeted covering space $X_1 \cong X_2$ , but such that there is no space having both $X_1$ and $X_2$ as covering spaces.

I tried thinking on some pairs of graphs but each pair I thought of, whom had an homeomorphic covering space, seemed to "inherit" the covering space's properties in a sense that I could always find a space which is covered by both graphs.
any suggestions?

Nathan Sikora
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    http://math.stackexchange.com/questions/1129047/partial-converse-of-existence-of-covering-spaces ... look at the solution proposed by allen hatcher ... :) – Anubhav Mukherjee Jul 06 '15 at 06:19

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