Randomized Greedy Online Edge Coloring Succeeds for Dense and Randomly-Ordered Graphs
Aditi Dudeja, Rashmika Goswami, and Michael Saks

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 can be properly edge colored with at most colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any graph on vertices with maximum degree using at most 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 -color the graph for arbitrary 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., . Prior to this work, the random greedy algorithm was only known to succeed in trees. Our second result is applicable even when the…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Complexity and Algorithms in Graphs
