  Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured that
an online $(1+o(1))\Delta$-edge-coloring algorithm exists for $n$-node graphs
of maximum degree $\Delta=\omega(\log n)$. This conjecture remains open in
general, though it was recently proven for bipartite graphs under
\emph{one-sided vertex arrivals} by Cohen et al.~[FOCS'19]. In a similar vein,
we study edge coloring under widely-studied relaxations of the online model.
  Our main result is in the \emph{random-order} online model. For this model,
known results fall short of the Bar-Noy et al.~conjecture, either in the degree
bound [Aggarwal et al.~FOCS'03], or number of colors used [Bahmani et
al.~SODA'10]. We achieve the best of both worlds, thus resolving the Bar-Noy et
al.~conjecture in the affirmative for this model.
  Our second result is in the adversarial online (and dynamic) model with
\emph{recourse}. A recent algorithm of Duan et al.~[SODA'19] yields a
$(1+\epsilon)\Delta$-edge-coloring with poly$(\log n/\epsilon)$ recourse. We
achieve the same with poly$(1/\epsilon)$ recourse, thus removing all dependence
on $n$.
  Underlying our results is one common offline algorithm, which we show how to
implement in these two online models. Our algorithm, based on the R\"odl Nibble
Method, is an adaptation of the distributed algorithm of Dubhashi et
al.~[TCS'98]. The Nibble Method has proven successful for distributed edge
coloring. We display its usefulness in the context of online algorithms.
