Improve worst case time of depth first search on Euler graphs
If I understand correctly, the only way to have a numerically ordered eulerian path would be having a path looking like 1,2,3..,n,1. All nodes in the path must have degree 2 (since you can only get into the once), except perhaps for nodes with degree 0, that you can instead skip.

If this is the case, the algorithm would just be a matter of testing if the 1,2,3..n path is eulerian, with no need to find eulerian paths with a traditional algorithm.