  Online Contention Resolution Schemes (OCRS's) represent a modern tool for
selecting a subset of elements, subject to resource constraints, when the
elements are presented to the algorithm sequentially. OCRS's have led to some
of the best-known competitive ratio guarantees for online resource allocation
problems, with the added benefit of treating different online decisions --
accept/reject, probing, pricing -- in a unified manner. This paper analyzes
OCRS's for resource constraints defined by matchings in graphs, a fundamental
structure in combinatorial optimization. We consider two dimensions of
variants: the elements being presented in adversarial or random order; and the
graph being bipartite or general. We improve the state of the art for all
combinations of variants, both in terms of algorithmic guarantees and
impossibility results. Some of our algorithmic guarantees are best-known even
compared to Contention Resolution Schemes that can choose the order of arrival
or are offline. All in all, our results for OCRS directly improve the
best-known competitive ratios for online accept/reject, probing, and pricing
problems on graphs in a unified manner.
