The Grundy (or First-Fit) chromatic number of a graph $G=(V,E)$, denoted by $\Gamma(G)$ (or $\chi_{_{\sf FF}}(G)$), is the maximum number of colors used by a First-Fit (greedy) coloring of $G$. To determine $\Gamma(G)$ is NP-complete for various classes of graphs. Also there exists a constant $c>0$ such that the Grundy number is hard to approximate within the ratio $c$. We first obtain an $\mathcal{O}(VE)$ algorithm to determine the Grundy number of block graphs i.e. graphs in which every biconnected component is complete subgraph. We prove that the Grundy number of a general graph $G$ with cut-vertices is upper bounded by the Grundy number of a block graph corresponding to $G$. This provides a reasonable upper bound for the Grundy number of graphs with cut-vertices. Next, define $\Delta_2(G)={\max}_{u\in G}~ {\max}_{v\in N(u):d(v)\leq d(u)} d(v)$. We obtain an $\mathcal{O}(VE)$ algorithm to determine $\Gamma(G)$ for graphs $G$ whose girth $g$ is at least $2\Delta_2(G)+1$. This algorithm provides a polynomial time approximation algorithm within ratio $\min \{1, (g+1)/(2\Delta_2(G)+2)\}$ for $\Gamma(G)$ of general graphs $G$ with girth $g$.

翻译：图$G=(V,E)$的Grundy（或称首次适应）色数，记作$\Gamma(G)$（或$\chi_{_{\sf FF}}(G)$），是指对$G$进行首次适应（贪心）着色时所用颜色的最大数目。对于许多图类，确定$\Gamma(G)$是NP完全问题。此外，存在常数$c>0$使得Grundy数在比值$c$内难以逼近。我们首先提出一种$\mathcal{O}(VE)$算法来确定块图（即每个双连通分量都是完全子图的图）的Grundy数。我们证明了一般含割点图$G$的Grundy数上界可由对应于$G$的某个块图的Grundy数界定，这为含割点图的Grundy数提供了一个合理的上界。其次，定义$\Delta_2(G)={\max}_{u\in G}~ {\max}_{v\in N(u):d(v)\leq d(u)} d(v)$。我们得到一种$\mathcal{O}(VE)$算法，用于确定围长$g$至少为$2\Delta_2(G)+1$的图$G$的$\Gamma(G)$。该算法为围长为$g$的一般图$G$的$\Gamma(G)$提供了一个比值在$\min \{1, (g+1)/(2\Delta_2(G)+2)\}$范围内的多项式时间逼近算法。