轻度伪随机正则图的哈密顿性
Hamiltonicity of mildly pseudorandom regular graphs
- University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明满足温和谱间隙条件的轻度伪随机正则图具有哈密顿性，通过随机矩阵不等式保留谱间隙以构造完美匹配，再用排序网络变体转化为哈密顿圈，给出更短证明和更优定量界。
AI中文摘要：
我们证明，若一个$(n,d,\lambda)$-图满足$\lambda\leq (1-\delta)d$且$d\gg \delta^{-6}(\log n)^{3}$（其中$\delta>0$），则该图是哈密顿的。Bradač和Janzer最近证明了性质上类似的结果。我们的证明更短，并给出了更好的定量界。在证明中，如同Ferber、Han、Mao和Vershynin的早期工作，我们使用一个随机矩阵不等式来表明，在随机抽取适当比例的顶点后，温和的谱间隙通常得以保留。这使我们能够推断，伪随机图的典型平衡二部子图包含完美匹配。为了将一组完美匹配转化为哈密顿圈，我们使用了排序网络方法的一个变体。
英文摘要：
We show that if an $(n,d,λ)$-graph satisfies $λ\leq (1-δ)d$ and $d\gg δ^{-6}(\log n)^{3}$ for some $δ>0$, then it is Hamiltonian. A qualitatively similar result was recently proven by Bradač and Janzer. Our proof here is shorter and gives better quantitative bounds. \par In our proof, as in earlier work of Ferber, Han, Mao, and Vershynin, we use a random matrix inequality to show that a mild spectral gap is typically preserved after randomly sampling an appropriate proportion of the vertices. This allows us to deduce that typical balanced bipartite subgraphs of pseudorandom graphs contain perfect matchings. To convert a collection of perfect matchings into a Hamilton cycle, we use a variant of the sorting network method.
