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I just started reading about graph theory and have a question (which might be trivial). How many $(2n-1)$ edge colorings of $K_{2n}$ are there?

A vaguer question: can I write $K_{4n}= K_4 + K_4 +.....+K_4$ where each $K_4$ has a 3-edge coloring considered as a subgraph of $K_n$? More generally, what's a good reference for edge-colorings of complete graphs.

thanks

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