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Do you know some unexpected and interesting applications of these hamiltonian and eulerian cycles/paths? For example, I have recently learnt about string reconstruction as an eulerian path problem and I was suspised a bit.
So, do you know something similar else?
Euler tour can be used to find LCA. $$$\mathcal O(n)$$$ for construction of depth array and $$$\mathcal O(logn)$$$ for query.
I think you're referring to an Euler Tour on a tree, not a Eulerian Path.