We prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.

中文翻译：

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稀疏伪随机图中的近乎完美的团因子

我们证明，对于任何$t \ge 3$, 存在一个常数C=C(吨) > 0 使得任何d-常规的n-绝对值 λ 中第二大特征值满足的顶点图$\lambda \le c{d^{t - 1}}/{n^{t - 2}}$包含顶点不相交的副本ķ吨涵盖所有但最多${n^{1 - 1/(8{t^4})}}$顶点。这进一步支持了 Krivelevich、Sudakov 和 Szábo 的猜想（组合学24(2004), pp. 403–426)n,d, λ)-图n∈ 3ℕ 和$\lambda \le c{d^2}/n$对于一个适当小的绝对常数C> 0 包含三角形因子。我们的论点将线性规划的工具与概率技术相结合，并将它们应用于特定的加权设置。我们预计这种方法将适用于该领域的其他问题。