Every 4-connected simple graph of odd order is odd 3-edge-colorable with the assumption necessary, and every connected Eulerian graph of odd order has an edge whose removal yields an odd 2-edge-colorable graph.
Petruˇ sevski, Odd 4-edge-colorability of graphs, J
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Odd Edge Colorings of Graphs with Odd Order
Every 4-connected simple graph of odd order is odd 3-edge-colorable with the assumption necessary, and every connected Eulerian graph of odd order has an edge whose removal yields an odd 2-edge-colorable graph.