Erdős–Rényi随机图中的同态核相变阈值
Homomorphic-core phase transition threshold in Erdős--Rényi random graphs
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中文总结 AI 辅助
该研究确定了Erdős–Rényi随机图成为同态核的相变阈值，推导了子图同构问题的紧ETH下界。
中文摘要 AI 辅助
本手稿证明，从Erdős–Rényi模型G(n,p)中抽取的随机图G，当p=p(n)≤1/2且lim_{n→+∞}(np−log n−log log n)=+∞时，G是同态核，即从G到自身的每个同态都是自同构。这意味着对于几乎所有平均度为多项式的k顶点模式，子图同构问题存在基于ETH的紧下界。
英文摘要
It is shown in this manuscript that a random graph $G$ drawn from the Erdős--Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.
