  We give an online algorithm that with high probability computes a
$\left(\frac{e}{e-1} + o(1)\right)\Delta$ edge coloring on a graph $G$ with
maximum degree $\Delta = \omega(\log n)$ under online edge arrivals against
oblivious adversaries, making first progress on the conjecture of Bar-Noy,
Motwani, and Naor in this general setting. Our algorithm is based on reducing
to a matching problem on locally treelike graphs, and then applying a tree
recurrences based approach for arguing correlation decay.
