Petersen graph edge chromatic number

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Hi I keep on getting 3 for the edge chromatic number of the Petersen graph. But the Petersen graph has edge chromatic number of 4 and I don’t know how to do that. Can someone please show this by proving this.

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Every cubic bridgeless graph has a perfect matching, hence we can find a perfect matching for the peterson graph and color the matched edges in one color. You can check that n o matter which perfect matching you choose to remove, After you remove the matched edges, you get two disjoint cycles on 5 vertices each. We know an odd cycle needs 3 colors to properly color the edges. Hence we can color the peterson graph using at most 4 colors and not less. So it's edge chromatic number is 4. I hope I did it right.