Given a connected graph $G(V,E)$, an Euler path (on edges) exists iff $G$ has either 0 or exactly 2 odd-degree vertices.
Suppose that $G(V,E)$ has $2n$ odd vertices. Let $P_m=\{G_1,\cdots,G_m\}$ be an edge partition of $G$ into $m$ connected subgraphs. Are there any qualitative bounds on the minimum $m$ needed such that each $G_i$ has an Euler path? Is there a formal name for this? Are there any quantitative results? My difficulty is in seeing whether you can even partition the graph into connected subgraphs with exactly 0 or 2 odd-degree nodes.
Equivalently, this problem can be thought of as: what is the minimum number of edge-colors $m$ that you need to cover all of $G$ such that each subgraph of colors has an euler circuit.
Here's an example on the famous Konigsberg Bridge problem, where there are two subgraphs (black and orange) on which Eulerian paths exist:
