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Chv\\'atal and Thomassen (JCTB, 1978) established general bounds for $f(d)$, and also proved that $f(2)=6$ and $f(4)\\geq 12$. The works of both Kwok, Liu and West (JCTB, 2010) and Wang and Chen (JCTB, 2022) together determined $f(3)=9$. 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