Don't worry ... no homework to school. I would just like to make a program that finds an eulerian path through a labeled graph so that in each vertex of the graph will the count be > 0.
My suggestion for finding an eulerian path on a nonlabeled graph is DFS(depht-first search) with complexity issue O(N+M), where N is a number of vertexes and M is a number of edges of the graph.
But I'm not quite sure how to do it with a labeled graph ... I would suggest again DFS, but when returning back instead of printing out the path just storing the edges how they go and counting the labels. Don't have a better idea.