🤖 AI Summary
This work addresses the (Δ+1)-edge-coloring problem for bounded-degree graphs with maximum degree Δ = O(1), optimizing upon Vizing’s theorem guarantee. Methodologically, it introduces a greedy reconstruction strategy, combinatorial probabilistic analysis, and novel distributed algorithm design—marking the first application of the entropy compression method to construct explicit edge-coloring algorithms. Contributions include: (i) the first centralized deterministic linear-time O(n) algorithm, improving over the prior optimal O(n log n) bound; (ii) the first deterministic Õ(log⁵ n) and randomized O(log² n) algorithms in the LOCAL model, substantially outperforming all existing results; and (iii) a unified framework bridging centralized and distributed settings via entropy compression. These advances yield key breakthroughs both in asymptotic time complexity and in algorithmic methodology for edge coloring.
📝 Abstract
Vizing's theorem states that every graph $G$ of maximum degree $Delta$ can be properly edge-colored using $Delta + 1$ colors. The fastest currently known $(Delta+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(msqrt{n})$, where $n :=|V(G)|$ and $m :=|E(G)|$. We investigate the case when $Delta$ is constant, i.e., $Delta = O(1)$. In this regime, the runtime of Sinnamon's algorithm is $O(n^{3/2})$, which can be improved to $O(n log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O(n)$, which is obviously best possible. Prior to this work, no linear-time $(Delta+1)$-edge-coloring algorithm was known for any $Delta geq 4$. Using some of the same ideas, we also develop new algorithms for $(Delta+1)$-edge-coloring in the $mathsf{LOCAL}$ model of distributed computation. Namely, when $Delta$ is constant, we design a deterministic $mathsf{LOCAL}$ algorithm with running time $ ilde{O}(log^5 n)$ and a randomized $mathsf{LOCAL}$ algorithm with running time $O(log ^2 n)$. Although our focus is on the constant $Delta$ regime, our results remain interesting for $Delta$ up to $log^{o(1)} n$, since the dependence of their running time on $Delta$ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method.