A strong Pَsa Condition for a graphic list to be forcibly -connected

Let π be a graphic list ( d 1 , … , d n ) , where n ≥ 3 and d 1 ≤ ⋯ ≤ d n . If each realization of π is Z 3 -connected, then π is said to be forcibly Z 3 -connected. If d m ≥ m + 1 for 1 ≤ m < n − 1 2 and d m + 1 ≥ m + 1 for m = n − 1 2 , then π is said to satisfy the Pósa Condition. In this paper, we prove that if π satisfies the Pósa Condition with d 1 ≥ 3 and d 2 ≥ 4 , then π is forcibly Z 3 -connected. This answers a question posed by Yin and Zhang [J.H. Yin, Y. Zhang, Pósa-condition and nowhere-zero 3-flows, Discrete Math. 311 (2011) 897–907].