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- W3192388985 abstract "Given a graph G, let fG(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 Ct be the cycle with vertex set [1,t] and edge set {12,23,…,(t−1)t,t1}. Aharoni et al. (2019) conjectured that (i) fG(n,n−1)=n−1 for all graphs G∈D(2) and (ii) fCt(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≥13n2+449n. An ordered set I=(a1,a2,…,an) on Ct is called a 2-jump independent n-set of Ct if ai+1−ai=2(modt) for any 1≤i≤n−1. We also show that a collection of 2-jump independent n-sets F of Ct with |F|=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∈D(2) with ce(G)≤4, where ce(G) is the number of components of G isomorphic to cycles of even lengths." @default.
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- W3192388985 date "2022-08-01" @default.
- W3192388985 modified "2023-10-16" @default.
- W3192388985 title "Rainbow independent sets in graphs with maximum degree two" @default.
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- W3192388985 doi "https://doi.org/10.1016/j.dam.2022.04.016" @default.
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