We study the performance of sequential contention resolution and matching algorithms on random graphs with vanishing edge probabilities. When the edges of the graph are processed in an adversarially-chosen order, we derive a new OCRS that is $0.382$-selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption. Complementary to this positive result, we show that no OCRS can be more than $0.390$-selectable, significantly improving upon the upper bound of $0.428$ from the literature. We also derive negative results that are specialized to bipartite graphs or subfamilies of OCRSs. Meanwhile, when the edges of the graph are processed in a uniformly random order, we show that the simple greedy contention resolution scheme which accepts all active and feasible edges is $1/2$-selectable. This result is tight due to a known upper bound. We then show that when the algorithm can choose the processing order, a slight tweak to the random order---give each vertex a random priority and process edges in lexicographic order---results in a strictly better contention resolution scheme that is $1-\ln(2-1/e)\approx0.510$-selectable. Moreover, we show that this bound is tight over any sequential contention resolution scheme, even one which may adaptively choose the order in which it processes edges. This provides a separation from the $0.544$ upper bound for offline contention resolution implied by the classic result of Karp and Sipser. Our positive results also apply to online matching on $1$-uniform random graphs with vanishing (non-identical) edge probabilities, extending and unifying some results from the random graphs literature.

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