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arXiv 2106.08975math.CO
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长诱导路的简短证明

Short proofs for long induced paths

Nemanja Draganić, Stefan Glock, Michael Krivelevich

备注：
to appear in CPC
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摘要：

我们提出了一个修改的深度优先搜索算法，适合寻找长诱导路径。我们用它来给出以下结果的简单证明。我们表明，诱导的大小Ramsey数的路径满足$\hat{R}_{\mathrm{ind}}(P_n)\leq 5\cdot 10^7n$，从而给出了一个明确的常数的线性范围内，改善了以前的范围与一个大的常数从一个正则性引理参数由Haxell，Kohayakawa和Chauczak。我们还提供了一个绑定的$k$彩色版本，显示$\hat{R}_{\mathrm{ind}}^k(P_n)=O(k^3\log^4k)n$ .最后，我们提出了一个新的短证明的事实，即二项式随机图在超临界政权，$G(n,\frac{1+\varepsilon}{n})$，通常包含一个诱导路径的长度$Θ(\varepsilon^2) n$。

英文摘要：

We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies $\hat{R}_{\mathrm{ind}}(P_n)\leq 5\cdot 10^7n$, thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and Łuczak. We also provide a bound for the $k$-color version, showing that $\hat{R}_{\mathrm{ind}}^k(P_n)=O(k^3\log^4k)n$. Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,\frac{1+\varepsilon}{n})$, contains typically an induced path of length $Θ(\varepsilon^2) n$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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