Deterministic Simple $(Δ+\varepsilonα)$-Edge-Coloring in Near-Linear Time

TL;DR

The paper resolves the long-standing open problem of deterministic near-linear time edge-coloring for simple graphs by giving a deterministic $(1+\varepsilon)\Delta$-edge-coloring algorithm running in $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$ time, and extends the approach to graphs of bounded arboricity by achieving a $(\Delta+\varepsilon\alpha)$-edge-coloring in $O\left(\frac{m\log n}{\varepsilon^7}\right)$ deterministic time (with randomized variants also provided). The core technique is a novel two-way degree-splitting built on alternating-directions paths, enabling balanced splits that drive a recursive coloring while keeping color surplus small; this is complemented by forests-decomposition orientation to control arboricity-driven parameters. The authors demonstrate both deterministic and randomized trade-offs, with the depth of recursion and color surplus depending on whether the graph is general or has bounded arboricity, respectively. Collectively, the work advances near-linear-time graph edge-coloring and introduces new tools—two-way degree-splitting and AD-paths—that may have broader applicability in edge-coloring and related combinatorial problems.

Abstract

We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $Δ$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides $(Δ+1)$-edge-coloring in $O(m\cdot n)$ deterministic time. This running time was improved to $O\left(m\cdot\min\left\{Δ\cdot\log n,\sqrt{n}\right\}\right)$, and very recently to randomized $\tilde{O}\left(m\cdot n^{1/3}\right)$. A randomized $(1+\varepsilon)Δ$-edge-coloring algorithm can be computed in $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$ time, and for large values of $Δ$, this task requires randomized $O\left(\frac{m\cdot\log\varepsilon^{-1}}{\varepsilon^2}\right)$ time. It was however open if there exists a deterministic near-linear time algorithm for this basic problem. We devise a simple deterministic $(1+\varepsilon)Δ$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$. A randomized variant of our algorithm has running time $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdotΔ)))$. We also study edge-coloring of graphs with arboricity at most $α$. A randomized computation of $(Δ+1)$-edge-coloring requires $\tilde{O}\left(\min\{m\cdot\sqrt{n},m\cdotΔ\}\cdot\fracαΔ\right)$ time. Deterministically, this task can be done in $O\left(m\cdotα^7\cdot\log n\right)$ time. However, for large values of $α$, these algorithms require super-linear time. We devise a deterministic $(Δ+\varepsilonα)$-edge-coloring algorithm with running time $O\left(\frac{m\cdot\log n}{\varepsilon^7}\right)$. A randomized version of our algorithm requires $O\left(\frac{m\cdot\log n}{\varepsilon}\right)$ expected time. Our algorithm is based on a novel two-way degree-splitting, which we devise in this paper. We believe that this technique is of independent interest.

Figures (4)

• Figure 1: Procedure Degree-Splitting
• Figure 2: Procedure Edge-Coloring
• Figure 3: Alternating-directions paths.
• Figure 4: An odd-length cycle alternating-directions path $(v_0,v_1,v_2, v_0)$.

Theorems & Definitions (51)

• Definition 1: Eulerian graph
• Definition 2: Adjacent edges
• Definition 3: Proper edge-coloring
• Theorem 1: $(\Delta+1)$-edge-coloring
• Theorem 2: Procedure Degree-Splitting
• Claim 1: Maximum degree of recursive graphs
• proof
• Claim 2: Recursive graphs' number of edges
• proof
• Lemma 1: Proper edge-coloring