Randomized Greedy Online Edge Coloring Succeeds for Dense and Randomly-Ordered Graphs

TL;DR

This paper demonstrates that a simple randomized greedy algorithm can efficiently edge color dense and randomly ordered graphs with near-optimal colors, extending its known success beyond trees to more complex graph classes.

Contribution

It proves that the naive random greedy algorithm succeeds in dense and randomly ordered graphs, and shows a deterministic variant exists for dense graphs, improving previous bounds.

Findings

Random greedy algorithm achieves (1+ε)Δ coloring in random edge order.

Success extends to dense graphs with adversarial edge arrival.

Existence of a deterministic algorithm with near-optimal coloring for dense graphs.

Abstract

Vizing's theorem states that any graph of maximum degree $\Delta$ can be properly edge colored with at most $\Delta+1$ colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any graph on $n$ vertices with maximum degree $\Delta = \omega(\log n)$ using at most $(1+o(1))\Delta$ colors. Here we study the na\"{i}ve random greedy algorithm, which simply chooses a legal color uniformly at random for each edge upon arrival. We show that this algorithm can $(1+\epsilon)\Delta$-color the graph for arbitrary $\epsilon$ in two contexts: first, if the edges arrive in a uniformly random order, and second, if the edges arrive in an adversarial order but the graph is sufficiently dense, i.e., $n = O(\Delta)$. Prior to this work, the random greedy algorithm was only known to succeed in trees. Our second result is applicable even when the adversary is adaptive, and therefore implies the existence of a deterministic edge coloring algorithm which $(1+\epsilon)\Delta$ edge colors a dense graph. Prior to this, the best known deterministic algorithm for this problem was the simple greedy algorithm which utilized $2\Delta-1$ colors.