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I just learned these theorems and I'm wondering how they are used? is it a guaranteed Hamilton cycle if either of these work? or do they both have to work? I'm wondering because I'm not sure if I should spend time proving it satisfies both theorems if I only need to test one of them

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Yes, it guarantees existance of Hamiltonian cycle. Moreover proof of both theorems are constructive, i. e., it gives polynomial-time algorithm for finding a Hamiltonian cycle, it graph satisfies appropriate degree condition.