$\mathcal{O}(VE)$ time algorithms for the Grundy (First-Fit) chromatic number of block graphs and graphs with sufficiently large girth

TL;DR

The paper addresses the computational challenge of the Grundy (First-Fit) chromatic number $Γ(G)$ for two graph families: block graphs and graphs with large girth. It develops a block-graph framework with the GRUNDY-BLOCK algorithm to compute $Γ(G)$ and per-cut-vertex values in near-linear time per cut-vertex, and it derives general bounds via the block-cutpoint structure, including $Γ(G)≤\tilde{Δ}(ω-1)+1$. For graphs with girth $g≥2Δ_2(G)+1$, it obtains a polynomial-time $O(nm)$ algorithm based on locality to subgraphs $G(v)$, plus an $O(nm)$-time approximation with a provable ratio, and a locality property $Γ(G)=\max_v Γ(G(v))$. These results yield exact and approximation techniques for important graph classes and illuminate the locality of Grundy colorings, with potential implications for online coloring and structure-aware heuristics.

Abstract

The Grundy (or First-Fit) chromatic number of a graph $G=(V,E)$, denoted by $Γ(G)$ (or $χ_{_{\sf FF}}(G)$), is the maximum number of colors used by a First-Fit (greedy) coloring of $G$. To determine $Γ(G)$ 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 $Δ_2(G)={\max}_{u\in G}~ {\max}_{v\in N(u):d(v)\leq d(u)} d(v)$. We obtain an $\mathcal{O}(VE)$ algorithm to determine $Γ(G)$ for graphs $G$ whose girth $g$ is at least $2Δ_2(G)+1$. This algorithm provides a polynomial time approximation algorithm within ratio $\min \{1, (g+1)/(2Δ_2(G)+2)\}$ for $Γ(G)$ of general graphs $G$ with girth $g$.

Figures (2)

• Figure 1: Assignment of lists to $(K_4,u)$ (left) and ASSIGN-LIST$(G,u)$ assigns the list $1234$ to $u$ using the lists $1, 1, 12, 12, 123$ in the neighborhood of $u$ (right)
• Figure 2: A full execution of GRUNDY-BLOCK$(G,w)$