Assume that some terminal component, Q , in D is a k-regular tournament ,

this implies that Q is also a terminal component in D, a contradiction Therefore no terminal component in D is a k-regular tournament and by induction there is a k-all-out-degreereducing 2k-partition of D . Now add w to a different part to all of its at most 2k ? 1 neighbours in G. This gives a k-all-out-degree-reducing 2k-partition of D ,

the graph G is not an odd cycle. Therefore, by Brook's Theorem, G admits a proper 2k-colouring. This 2k-colouring gives us the desired k-all-out-degree-reducing 2k-partition of D ,

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