Let $\Sigma$ be a signed graph where two edges joining the same pair of vertices with opposite signs are allowed. The zero-free chromatic number $\chi^*(\Sigma)$ of $\Sigma$ is the minimum even integer $2k$ such that $G$ admits a proper coloring $f\colon\,V(\Sigma)\mapsto \{\pm 1,\pm 2,\ldots,\pm k\}$. The zero-free list chromatic number $\chi^*_l(\Sigma)$ is the list version of zero-free chromatic number. $\Sigma$ is called zero-free chromatic-choosable if $\chi^*_l(\Sigma)=\chi^*(\Sigma)$. We show that if $\Sigma$ has at most $\chi^*(\Sigma)+1$ vertices then $\Sigma$ is zero-free chromatic-choosable. This result strengthens Noel-Reed-Wu Theorem which states that every graph $G$ with at most $2\chi(G)+1$ vertices is chromatic-choosable, where $\chi(G)$ is the chromatic number of $G$.

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Noel-Reed-Wu 定理对有符号图的推广

令 $\Sigma$ 是一个有符号图，其中允许连接同一对具有相反符号的顶点的两条边。$\Sigma$ 的无零色数 $\chi^*(\Sigma)$ 是最小偶数 $2k$，使得 $G$ 允许正确着色 $f\colon\,V(\Sigma)\映射到 \{\pm 1,\pm 2,\ldots,\pm k\}$。零自由色数 $\chi^*_l(\Sigma)$ 是零自由色数的列表版本。如果 $\chi^*_l(\Sigma)=\chi^*(\Sigma)$，则 $\Sigma$ 被称为零自由色选择。我们证明如果 $\Sigma$ 至多有 $\chi^*(\Sigma)+1$ 个顶点，那么 $\Sigma$ 是零自由色选择的。这个结果加强了 Noel-Reed-Wu 定理，该定理指出每个图 $G$ 最多具有 $2\chi(G)+1$ 个顶点是色可选的，其中 $\chi(G)$ 是 $G 的色数$.