O(VE) time algorithms for the Grundy (First-Fit) chromatic number of block graphs and graphs with large girth

The Grundy (or First-Fit) chromatic number of a graph $G=(V,E)$, denoted by $\Gamma \left(G\right)$ (or ${\chi }_{{}_{\mathrm{FF}}}\left(G\right)$), is the maximum number of colors used by a First-Fit (greedy) coloring of G. The determining $\Gamma \left(G\right)$ 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 $O\left(VE\right)$ algorithm to determine the Grundy number of block graphs i.e. graphs in which every biconnected component is a complete graph. 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 ${\Delta }_2\left(G\right)={\max }_{u\in V}{\max }_{v\in N\left(u\right):d\left(v\right)\le d\left(u\right)}d\left(v\right)$. We obtain an $O\left(VE\right)$ algorithm to determine $\Gamma \left(G\right)$ for graphs G whose girth g is at least $2{\Delta }_2\left(G\right)+1$. This algorithm provides a polynomial time approximation algorithm within ratio $\min \{1,(g+1)/(2{\Delta }_2\left(G\right)+2\left)\right\}$ for $\Gamma \left(G\right)$ of general graphs G with girth g.