Randomly keeping each edge with probability about 1/d still leaves a cycle of length nearly d in any average-degree-d graph, and graphs with no d-cycle are nearly disjoint unions of components with small vertex covers.
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Robustness and hyperstability for the Erd\H{o}s-Gallai theorem
Randomly keeping each edge with probability about 1/d still leaves a cycle of length nearly d in any average-degree-d graph, and graphs with no d-cycle are nearly disjoint unions of components with small vertex covers.