Alexander Souza has offered for review on arXiv a constructive solution in the affirmative.
http://arxiv.org/abs/1202.0569
Edit: Unless we are mistaken, the Petersen graph is a counterexample to Theorem 2. Furthermore there is an unresolved problem with Lemma 9.
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Claimed solution in the affirmative
Alexander Souza has offered for review on arXiv a constructive solution in the affirmative.
http://arxiv.org/abs/1202.0569
Edit: Unless we are mistaken, the Petersen graph is a counterexample to Theorem 2. Furthermore there is an unresolved problem with Lemma 9.