Rainbow independent sets in graphs with maximum degree two

TL;DR

This paper investigates rainbow independent sets in graphs with maximum degree two, proving conjectures for large cycle lengths and specific families of independent sets, and establishing connections between different conjectures.

Contribution

It proves the conjecture for cycle lengths beyond a quadratic threshold and explores conditions under which the conjecture implies another related conjecture.

Findings

Conjecture (ii) holds for cycle length t ≥ (1/3)n^2 + (44/9)n.

A collection of 2-jump independent n-sets admits a rainbow independent n-set.

If conjecture (ii) holds, then conjecture (i) holds for graphs with up to 4 even cycle components.

Abstract

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.