  We study online bipartite edge coloring, with nodes on one side of the graph
revealed sequentially. The trivial greedy algorithm is $(2-o(1))$-competitive,
which is optimal for graphs of low maximum degree, $\Delta=O(\log n)$ [BNMN
IPL'92]. Numerous online edge-coloring algorithms outperforming the greedy
algorithm in various settings were designed over the years (e.g., AGKM FOCS'03,
BMM SODA'10, CPW FOCS'19, BGW SODA'21, KLSST STOC'22, BSVW STOC'24), all
crucially relying on randomization. A commonly-held belief, first stated by
[BNMN IPL'92], is that randomization is necessary to outperform greedy.
  Surprisingly, we refute this belief, by presenting a deterministic algorithm
that beats greedy for sufficiently large $\Delta=\Omega(\log n)$, and in
particular has competitive ratio $\frac{e}{e-1}+o(1)$ for all
$\Delta=\omega(\log n)$. We obtain our result via a new and surprisingly simple
randomized algorithm that works against adaptive adversaries (as opposed to
oblivious adversaries assumed by prior work), which implies the existence of a
similarly-competitive deterministic algorithm [BDBKTW STOC'90].
