Deterministic Simple (1+ε)Δ-Edge-Coloring in Near-Linear Time

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· n) deterministic time. This running time was improved to O(m·min{Δ·log n, √(n)}). It is also well-known that 3⌈Δ/2⌉-edge-coloring can be computed in O(m·logΔ) time deterministically. Duan et al. devised a randomized (1+ε)Δ-edge-coloring algorithm with running time O(m·log^6 n/ε^2). It was however open if there exists a deterministic near-linear time algorithm for this basic problem. We devise a simple deterministic (1+ε)Δ-edge-coloring algorithm with running time O(m·log n/ε). We also devise a randomized (1+ε)Δ-edge-coloring algorithm with running time O(m·(ε^-18+log(ε·Δ))). For ε≥1/log^1/18Δ, this running time is O(m·logΔ).