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arXiv 2108.02520math.CO
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最大度为2的图中的彩虹独立集

Rainbow independent sets in graphs with maximum degree two

Yue Ma, Xinmin Hou, Jun Gao, Boyuan Liu, Zhi Yin

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摘要：

给定一个图$G$，设$f_{G}(n,m)$是使$G$中每个$k$独立的$n$ -集都有彩虹$m$ -集的最小数.设$\mathcal{D}(2)$是所有最大度不超过2的图的族。Aharoni等人（2019）证实了（i）所有图$G\in\mathcal{D}(2)$的$f_G(n,n-1)=n-1$和（ii）$t\ge 2n+1$的$f_{C_t}(n,n)=n$。Lv和Lu（2020）证明了当$t=2n+1$时猜想（ii）成立。在这篇文章中，我们证明了猜想（ii）对$t\ge\frac{1}{3}n^2+\frac{44}{9}n$成立。令$C_t$是长度为$t$的循环，其顶点按顺时针顺序排列。$C_t$上的有序集$I=(a_1,a_2,\ldots,a_n)$称为$C_t$的$2$ -跳独立$n$ -集，如果对任何$1\le i\le n-1$有$a_{i+1}-a_i=2\pmod{t}$.我们还表明，收集2-跳跃独立的$n$ -集$\mathcal{F}$的$C_t$与$|\mathcal{F}|=n$承认彩虹独立的$n$ -集，即（ii）举行，如果我们限制$\mathcal{F}$的家庭2-跳跃独立的$n$ -集。此外，我们证明了如果猜想（ii）成立，则猜想（i）对所有具有$c_e(G)\le 4$的图$G\in\mathcal{D}(2)$成立，其中$c_e(G)$是$G$同构于偶数长圈的分支数.

英文摘要：

Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $\mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019) conjectured that (i) $f_G(n,n-1)=n-1$ for all graphs $G\in\mathcal{D}(2)$ and (ii) $f_{C_t}(n,n)=n$ for $t\ge 2n+1$. Lv and Lu (2020) showed that the conjecture (ii) holds when $t=2n+1$. In this article, we show that the conjecture (ii) holds for $t\ge\frac{1}{3}n^2+\frac{44}{9}n$. Let $C_t$ be a cycle of length $t$ with vertices being arranged in a clockwise order. An ordered set $I=(a_1,a_2,\ldots,a_n)$ on $C_t$ is called a $2$-jump independent $n$-set of $C_t$ if $a_{i+1}-a_i=2\pmod{t}$ for any $1\le i\le n-1$. We also show that a collection of 2-jump independent $n$-sets $\mathcal{F}$ of $C_t$ with $|\mathcal{F}|=n$ admits a rainbow independent $n$-set, i.e. (ii) holds if we restrict $\mathcal{F}$ on the family of 2-jump independent $n$-sets. Moreover, we prove that if the conjecture (ii) holds, then (i) holds for all graphs $G\in\mathcal{D}(2)$ with $c_e(G)\le 4$, where $c_e(G)$ is the number of components of $G$ isomorphic to cycles of even lengths.

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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