A conflict-free open neighborhood coloring of a graph is an assignment of colors to the vertices such that for every vertex there is a color that appears exactly once in its open neighborhood. For a graph G, the smallest number of colors required for such a coloring is called the conflict-free open neighborhood (CFON) chromatic number and is denoted by \chi_{ON}(G). Analogously, we define conflict-free closed neighborhood (CFCN) coloring, and CFCN chromatic number (denoted by \chi_{CN}(G)). First studied in 2002, this problem has received considerable attention. We study the CFON and CFCN coloring problems and show the following results. In what follows, \Delta denotes the maximum degree of the graph. 1. For a K_{1, k}-free graph G, we show that \chi_{ON}(G) = O(k \ln\Delta). This improves the bound of O(k^2 \ln \Delta) from [Bhyravarapu, Kalyanasundaram, Mathew, MFCS 2022]. Since \chi_{CN}(G) \leq 2\chi_{ON}(G), our result implies an upper bound on \chi_{CN}(G) as well. It is known that there exist separate classes of graphs with \chi_{ON}(G) = \Omega(\ln\Delta) and \chi_{ON}(G) = \Omega(k). 2. Let f(\delta) be defined as follows: f(\delta) = max {\chi_{CN} (G) : G is a graph with minimum degree \delta}. It is easy to see that f(\delta') \geq f(\delta) when \delta' < \delta. It is known [Debski and Przybylo, JGT 2021] that f(c \Delta) = \Theta(\log \Delta), for any positive constant c. In this paper, we show that f(c\Delta^{1 - \epsilon}) = \Omega (\ln^2 \Delta), where c, \epsilon are positive constants such that \epsilon < 0.75. Together with the known upper bound \chi_{CN}(G) = O(\ln^2 \Delta), this implies that f(c\Delta^{1 - \epsilon}) = \Theta (\ln^2 \Delta). 3. For a K_{1, k}-free graph G on n vertices, we show that \chi_{CN}(G) = O(\ln k \ln n). This bound is asymptotically tight.

翻译：图的冲突自由开邻域染色是对顶点的一种着色，使得每个顶点的开邻域中存在一种颜色恰好出现一次。对于图G，实现这种染色所需的最小颜色数称为冲突自由开邻域（CFON）色数，记为χ_ON(G)。类似地，可以定义冲突自由闭邻域（CFCN）染色及其色数（记为χ_CN(G)）。该问题自2002年首次研究以来已受到广泛关注。本文研究CFON和CFCN染色问题，得到以下结果（其中Δ表示图的最大度）：1. 对于K_{1, k}-自由图G，证明χ_ON(G) = O(k lnΔ)，改进了[Bhyravarapu, Kalyanasundaram, Mathew, MFCS 2022]中的界O(k^2 ln Δ)。由于χ_CN(G) ≤ 2χ_ON(G)，该结果也给出了χ_CN(G)的上界。已知存在图类分别满足χ_ON(G) = Ω(lnΔ)和χ_ON(G) = Ω(k)。2. 定义f(δ) = max {χ_CN(G): 图G的最小度为δ}。易见当δ' < δ时f(δ') ≥ f(δ)。已知[Debski and Przybylo, JGT 2021]对于任意正常数c，有f(cΔ) = Θ(logΔ)。本文证明对于正常数c, ε满足ε < 0.75，有f(cΔ^{1 - ε}) = Ω(ln^2 Δ)。结合已知上界χ_CN(G) = O(ln^2 Δ)，可得f(cΔ^{1 - ε}) = Θ(ln^2 Δ)。3. 对于n阶K_{1, k}-自由图G，证明χ_CN(G) = O(ln k ln n)，该界渐近紧确。