A proper $k$-coloring of a graph $G$ is a \emph{neighbor-locating $k$-coloring} if for each pair of vertices in the same color class, the two sets of colors found in their respective neighborhoods are different. The \textit{neighbor-locating chromatic number} $\chi_{NL}(G)$ is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. A proper $k$-vertex-coloring of a graph $G$ is a \emph{locating $k$-coloring} if for each pair of vertices $x$ and $y$ in the same color-class, there exists a color class $S_i$ such that $d(x,S_i)\neq d(y,S_i)$. The locating chromatic number $\chi_{L}(G)$ is the minimum $k$ for which $G$ admits a locating $k$-coloring. Our main results concern the largest possible order of a sparse graph of given neighbor-locating chromatic number. More precisely, we prove that if $G$ has order $n$, neighbor-locating chromatic number $k$ and average degree at most $2a$, where $2a\le k-1$ is a positive integer, then $n$ is upper-bounded by $\mathcal{O}(a^2(k^{2a+1}))$. We also design a family of graphs of bounded maximum degree whose order is close to reaching this upper bound. Our upper bound generalizes two previous bounds from the literature, which were obtained for graphs of bounded maximum degree and graphs of bounded cycle rank, respectively. Also, we prove that determining whether $\chi_L(G)\le k$ and $\chi_{NL}(G)\le k$ are NP-complete for sparse graphs: more precisely, for graphs with average degree at most 7, maximum average degree at most 20 and that are $4$-partite. We also study the possible relation between the ordinary chromatic number, the locating chromatic number and the neighbor-locating chromatic number of a graph.

翻译：图$G$的一个正常$k$-染色若满足：对于同一颜色类中的任意两个顶点，它们各自邻域内所包含的颜色集合互不相同，则称为\emph{邻域定位$k$-染色}。\textit{邻域定位色数} $\chi_{NL}(G)$ 定义为使得$G$存在邻域定位$k$-染色的最小$k$值。图$G$的一个正常$k$-顶点染色若满足：对于同一颜色类中的任意两个顶点$x$和$y$，存在一个颜色类$S_i$使得$d(x,S_i)\neq d(y,S_i)$，则称为\emph{定位$k$-染色}。\textit{定位色数} $\chi_{L}(G)$ 定义为使得$G$存在定位$k$-染色的最小$k$值。我们的主要结果涉及给定邻域定位色数的稀疏图的最大可能阶数。具体而言，我们证明：若图$G$的阶数为$n$，邻域定位色数为$k$，且平均度数至多为$2a$（其中$2a\le k-1$为正整数），则$n$的上界为$\mathcal{O}(a^2(k^{2a+1}))$。我们还设计了一个具有有界最大度数的图族，其阶数接近该上界。该上界推广了文献中两个分别针对有界最大度数图和界循环秩图的结果。此外，我们证明对于稀疏图而言，判定$\chi_L(G)\le k$与$\chi_{NL}(G)\le k$是NP完全的——具体来说，对于平均度数不超过7、最大平均度数不超过20且为4-部图的图类成立。我们还研究了图的普通色数、定位色数与邻域定位色数之间可能存在的关联。