G. If, G. Bipartite-then, =. G+g, and . So, Then Proposition 3

H. =. Set, By Proposition Let us now show that H × K 2 contains a T k+3 By construction, the subgraph of G ? × K 2 induced by (V (T k ) \ (S × K 2 )) ? S v?S {(v 1 , 1), (v 2 , 2)} is a T k . Note that for every vertex v ? V (G ? ) at most one of {(v, 1), (v, 2)} is in V (T ? ) Now every vertex of T ? is the root of a T 3 in its associated J × K 2 . All these T 3 together with T ? form an induced T k+3

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