Deterministic Simple $(\Delta+\varepsilon\alpha)$-Edge-Coloring in Near-Linear Time

We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $\Delta$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides $(\Delta+1)$-edge-coloring in $O(m\cdot n)$ deterministic time. This running time was improved to $O\left(m\cdot\min\left\{\Delta\cdot\log n,\sqrt{n}\right\}\right)$, and very recently to randomized $\tilde{O}\left(m\cdot n^{1/3}\right)$. A randomized $(1+\varepsilon)\Delta$-edge-coloring algorithm can be computed in $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$ time, and for large values of $\Delta$, 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 $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$. A randomized variant of our algorithm has running time $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdot\Delta)))$. We also study edge-coloring of graphs with arboricity at most $\alpha$. A randomized computation of $(\Delta+1)$-edge-coloring requires $\tilde{O}\left(\min\{m\cdot\sqrt{n},m\cdot\Delta\}\cdot\frac{\alpha}{\Delta}\right)$ time. Deterministically, this task can be done in $O\left(m\cdot\alpha^7\cdot\log n\right)$ time. However, for large values of $\alpha$, these algorithms require super-linear time. We devise a deterministic $(\Delta+\varepsilon\alpha)$-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.