On the strict Chvátal-condition and nowhere-zero 3-flows

Let G be a simple graph on $n\ge 3$ vertices and $(d_1,\dots ,d_n)$ be the degree sequence of G with $d_1\le \cdots \le d_n$. The classical Chvátal's theorem states that if $d_j\ge j+1$ or $d_{n-j}\ge n-j$ for each j with $1\le j<\frac{n}{2}$, then G is hamiltonian, which implies that G has a nowhere-zero 4-flow. Given an integer i with $2\le i<\frac{n}{2}$, we say that the graph G satisfies the strict Chvátal-condition on i if $d_j\ge j+1$ for each $j\ne i$ with $1\le j<\frac{n}{2}$, $d_i=i$ and $d_{n-i}\ge n-i$. In this paper, we show that if G satisfies the strict Chvátal-condition on i for some i with $2\le i<\frac{n}{2}$, then G has no nowhere-zero 3-flow if and only if $i=3$ and $G\in \{G_8,G_9\}$ as described in Fig. 2.