The widespread application of graph data inevitably brings significant privacy risks, as its unprotected use can lead to the leakage of sensitive information. These risks are particularly acute in the setting with an untrusted curator, where the data remains decentralized and each user only holds the connections to their neighbors. To mitigate such privacy risks, we adopt local differential privacy (LDP) to collect users' private information and generate a synthetic graph. However, existing methods suffer from either excessive noise injection by perturbing the local adjacency lists or significant structural information loss due to the simplistic graph encoding process. To address these issues, we propose LDPGraph, an effective graph synthesis algorithm that takes one step further by exploiting neighborhood structures under LDP. To obtain neighborhood statistics beyond degrees, LDPGraph aggregates a noisy global view from perturbed adjacency lists and combines it with projected local connections to estimate node-level triangle counts. To correct the structural inconsistency caused by separately perturbing degree and triangle count, LDPGraph jointly refines them into feasible degree-triangle targets. To reconstruct a global graph from these estimated targets, LDPGraph adopts a triangle-first strategy that first preserves local clustering structures and then fulfills remaining degree requirements. Extensive experiments on four real-world datasets and multiple commonly used graph metrics validate the superiority of LDPGraph. Source code is available at https://github.com/ZJU-TrustAID/LDPGraph.

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