随机图的高效哈密顿覆盖与线性荫度
Efficient Hamilton covers and linear arboricity of random graphs
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中文总结 AI 辅助
研究随机图$G(n,p)$的哈密顿覆盖，在广泛边概率范围证明其大小下界紧性，尤其在稀疏区域有新贡献，开发工具分解图并扩展为哈密顿圈，还证明击中时间结果，且表明该随机图对各$p\leq 1$满足线性荫度猜想。
中文摘要 AI 辅助
图的哈密顿覆盖是一组哈密顿圈，其并集包含所有边。由于每个哈密顿圈在每个顶点覆盖两条边，所以每个哈密顿覆盖的大小至少为$\lceil \Delta(G)/2\rceil$。我们证明，在最广泛的边概率范围内，对于二项随机图$G(n,p)$，这个下界是紧的：若$\omega(n)\to\infty$且\(\frac{\log n+\log\log n+\omega(n)}{n} \le p=p(n) \le 1-\frac{\omega(n)}{n^{2}}\)，则$G\sim G(n,p)$以高概率有大小为\(\left\lceil \frac{\Delta(G)}{2}\right\rceil\)的哈密顿覆盖。主要新贡献在于哈密顿性阈值附近的稀疏区域，证明了Draganić等人的一个猜想。我们的证明开发了构造性工具，将此类图分解为受控森林系统并扩展为哈密顿圈。还证明了随机图过程的相应击中时间结果。最后，用方法表明对于每个$p\leq 1$，$G\sim G(n,p)$以高概率满足著名的线性荫度猜想。
英文摘要
A Hamilton cover of a graph is a collection of Hamilton cycles whose union contains all edges. Since each Hamilton cycle covers two edges at every vertex, every Hamilton cover has size at least $\lceil Δ(G)/2\rceil$. We prove that this lower bound is tight for binomial random graphs $G(n,p)$ throughout the widest possible range of edge probabilities: if $ω(n)\to\infty$ and \[ \frac{\log n+\log\log n+ω(n)}{n} \le p=p(n) \le 1-\frac{ω(n)}{n^{2}}, \] then $G\sim G(n,p)$ with high probability has a Hamilton cover of size $\left\lceil \frac{Δ(G)}{2}\right\rceil. $ The main new contribution is the sparse regime near the Hamiltonicity threshold, where we prove a conjecture of Draganić, Glock, Munhá Correia and Sudakov. Our proof develops constructive tools for decomposing such graphs into controlled forest systems and extending them, using reserved pseudorandom structure, into Hamilton cycles. We also prove the corresponding hitting-time result for the random graph process, answering a question of Hefetz, Kühn, Lapinskas and Osthus. Finally, we use our methods to show that $G\sim G(n,p)$ with high probability satisfies the celebrated Linear arboricity conjecture for every $p\leq 1$.
