Density-Sensitive Algorithms for $(Δ+ 1)$-Edge Coloring
Sayan Bhattacharya, Martín Costa, Nadav Panski, Shay Solomon
TL;DR
This work tackles fast computation of a $(Δ+1)$-edge coloring for general graphs by introducing density-sensitive refinements guided by arboricity $α$. Building on Sinnamon’s randomized framework, the authors prioritize low-degree edges via the edge weight $w(e)=\min\{d(u),d(v)\}$ and develop a weight-aware analysis that bounds the total work to $O(mα)$ per recursion level. They integrate a refined Euler-partition divide-and-conquer (Partition, Recurse, Prune, Repair) and replace the repair step with a density-sensitive Color-Edges routine, achieving an overall randomized running time of $\tilde{O}((\min\{m\sqrt{n}, mΔ\})\cdot \frac{α}{Δ})$, with near-linear performance for bounded arboricity and favorable regimes when $α=\tilde{O}(Δ/\sqrt{n})$. The results extend prior barriers by factor $α/Δ$ and provide a framework that blends weight-based analysis, probabilistic coloring, and Euler partition techniques, improving practical performance on sparse graphs while preserving theoretical guarantees. This density-sensitive methodology has potential implications for fast graph coloring in sparse networks and related combinatorial optimization tasks.
Abstract
Vizing's theorem asserts the existence of a $(Δ+1)$-edge coloring for any graph $G$, where $Δ= Δ(G)$ denotes the maximum degree of $G$. Several polynomial time $(Δ+1)$-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})$, by Gabow et al.\ from 1985, where $n$ and $m$ denote the number of vertices and edges in the graph, respectively. (The $\tilde{O}$ notation suppresses polylogarithmic factors.) Recently, Sinnamon shaved off a polylogarithmic factor from the time bound of Gabow et al. The {arboricity} $α= α(G)$ of a graph $G$ is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph's "uniform density". While $α\le Δ$ in any graph, many natural and real-world graphs exhibit a significant separation between $α$ and $Δ$. In this work we design a $(Δ+1)$-edge coloring algorithm with a running time of $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})\cdot \fracαΔ$, thus improving the longstanding time barrier by a factor of $\fracαΔ$. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., $α= \tilde{O}(1)$) as well as when $α= \tilde{O}(\fracΔ{\sqrt{n}})$. Our algorithm builds on Sinnamon's algorithm, and can be viewed as a density-sensitive refinement of it.