Efficient Parallel $(Δ+1)$-Edge-Coloring

TL;DR

This work studies efficient parallel $(Δ+1)$-edge-coloring in the PRAM model, advancing beyond prior MIS-based schemes by introducing a parallel fan-recoloring framework built on a new fan-graph $G^{(F)}$ and large independent-set techniques. By combining a refined MIS-based fan construction with parallel recolorings and a Reduce-Color mechanism, the authors achieve running times of $O(Δ^4\log^4 n)$ with $O(mΔ)$ processors, plus variants with $O(Δ^{4+o(1)}\log^2 n)$ time under different MIS primitives and near-optimal tradeoffs for bounded arboricity. They also extend these ideas to $(1+ε)Δ$-edge-coloring, provide an improved edge-coloring update algorithm, and offer a detailed technical overview linking to the classical Karloff–Shmoys and Liang–Shen–Hu algorithms. The results yield faster, deterministic parallel colorings with broad applicability to sparse graphs and dynamic settings, significantly improving previous PRAM bounds and offering flexible time–processor tradeoffs. The paper contributes both theoretical insights and practical parallel approaches for fundamental graph coloring tasks.

Abstract

We study the $(Δ+1)$-edge-coloring problem in the parallel $\left(\mathrm{PRAM}\right)$ model of computation. The celebrated Vizing's theorem [Viz64] states that every simple graph $G = (V,E)$ can be properly $(Δ+1)$-edge-colored. In a seminal paper, Karloff and Shmoys [KS87] devised a parallel algorithm with time $O\left(Δ^5\cdot\log n\cdot\left(\log^3 n+Δ^2\right)\right)$ and $O(m\cdotΔ)$ processors. This result was improved by Liang et al. [LSH96] to time $O\left(Δ^{4.5}\cdot \log^3Δ\cdot \log n + Δ^4 \cdot\log^4 n\right)$ and $O\left(n\cdotΔ^{3} +n^2\right)$ processors. [LSH96] claimed $O\left(Δ^{3.5} \cdot\log^3Δ\cdot \log n + Δ^3\cdot \log^4 n\right)$ time, but we point out a flaw in their analysis, which once corrected, results in the above bound. We devise a faster parallel algorithm for this fundamental problem. Specifically, our algorithm uses $O\left(Δ^4\cdot \log^4 n\right)$ time and $O(m\cdot Δ)$ processors. Another variant of our algorithm requires $O\left(Δ^{4+o(1)}\cdot\log^2 n\right)$ time, and $O\left(m\cdotΔ\cdot\log n\cdot\log^δΔ\right)$ processors, for an arbitrarily small $δ>0$. We also devise a few other tradeoffs between the time and the number of processors, and devise an improved algorithm for graphs with small arboricity. On the way to these results, we also provide a very fast parallel algorithm for updating $(Δ+1)$-edge-coloring. Our algorithm for this problem is dramatically faster and simpler than the previous state-of-the-art algorithm (due to [LSH96]) for this problem.

Figures (14)

• Figure 1: In all figures, the color of each vertex $u_i\in\langle u_1,u_2,...,u_7\rangle$ represents a free (i.e., missing) color at $u_i$. An uncolored edge is represented by a dotted line. This figure depicts a maximal fan $\langle u_1,u_2,...,u_7\rangle$ centered at $v$ with missing colors $\color{violet}\bullet\color{black}$ and $\langle\color{blue}\bullet\color{black},\color{red}\bullet\color{black},\color{pink}\bullet\color{black},\color{olive}\bullet\color{black},\color{cyan}\bullet\color{black},\color{brown}\bullet\color{black},\color{pink}\bullet\color{black}\rangle$, that is characterized by $(\alpha,\beta)=(\color{violet}\bullet\color{black}, \color{pink}\bullet\color{black})$.
• Figure 2: The last step of the construction process of a maximal fan. In this example, $\alpha_k=\color{brown}\bullet\color{black}$, $\alpha_{k+1}=\alpha_3=\color{pink}\bullet\color{black}$ and $(v,u_4)$ is colored $\color{pink}\bullet\color{black}$ and is already in the fan. Also, $(v,u_{k+1})$ is colored $\color{brown}\bullet\color{black}$. This color did not appear in the fan $f$.
• Figure 3: Rotation of a fan $\langle u_1,u_2,...,u_7\rangle$ centered at $v$. In all figures, black color indicates a "wild-card", i.e., an undetermined missing color.
• Figure 4: $P$ is the $\alpha\beta$-path of a maximal fan $f$. The fan is characterized by $(\alpha,\beta)=(\color{violet}\bullet\color{black},\color{pink}\bullet\color{black})$.
• Figure 5: Two examples of a fan $\langle u_1,u_2,...,u_7\rangle$ centered at $v$ and characterized by $(\alpha,\beta)=(\color{violet}\bullet,\color{pink}\bullet\color{black})$ with its special vertices $x(v)$, $y(v)$ and $z(v)$, in the two possible cases.

Theorems & Definitions (131)

• Definition 1: Eulerian graph
• Definition 2: Adjacent edges
• Definition 3: Proper edge-coloring
• Definition 4: Arboricity
• Definition 5: Orientation
• Claim 1
• proof
• Definition 6: Degeneracy
• Claim 2: A bound on the degeneracy
• Definition 7: Fan