Rainbow independent sets in graphs with maximum degree two

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 D(2) be the family of all graphs with maximum degree at most two. For t >= 3, let C-t be the cycle with vertex set [1, t] and edge set {12, 23, ..., (t - 1)t, t1}. Aharoni et al. (2019) conjectured that (i) f(G)(n, n-1) = n - 1 for all graphs G is an element of D(2) and (ii) f(Ct)(n, n) = n for t >= 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 >= 1/3n(2) + 44/9n. An ordered set I = (a(1), a(2), , a(n)) on C-t is called a 2-jump independent n-set of C-t if a(i+1) a(i) = 2 (mod t) for any 1 <= i <= n - 1. We also show that a collection of 2-jump independent n-sets F of C-t with vertical bar F vertical bar = n admits a rainbow independent n-set, i.e. (ii) holds if we restrict 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 is an element of D(2) with ce(()G) <= 4, where c(e)(G) is the number of components of G isomorphic to cycles of even lengths. (C) 2022 Elsevier B.V. All rights reserved.