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 sets of colors found in their neighborhoods are different. The neighbor-locating chromatic number $\chi_{NL}(G)$ is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. A proper $k$-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. It follows that $\chi(G)\leq\chi_L(G)\leq\chi_{NL}(G)$ for any graph $G$, where $\chi(G)$ is the usual chromatic number of $G$. We show that for any three integers $p,q,r$ with $2\leq p\leq q\leq r$ (except when $2=p=q<r$), there exists a connected graph $G_{p,q,r}$ with $\chi(G_{p,q,r})=p$, $\chi_L(G_{p,q,r})=q$ and $\chi_{NL}(G_{p,q,r})=r$. We also show that the locating chromatic number (resp., neighbor-locating chromatic number) of an induced subgraph of a graph $G$ can be arbitrarily larger than that of $G$. Alcon \textit{et al.} showed that the number $n$ of vertices of $G$ is bounded above by $k(2^{k-1}-1)$, where $\chi_{NL}(G)=k$ and $G$ is connected (this bound is tight). When $G$ has maximum degree $\Delta$, they also showed that a smaller upper-bound on $n$ of order $k^{\Delta+1}$ holds. We generalize the latter by proving that if $G$ has order $n$ and at most $an+b$ edges, then $n$ is upper-bounded by a bound of the order of $k^{2a+1}+2b$. Moreover, we describe constructions of such graphs which are close to reaching the bound.

翻译：图$G$的一个正常$k$-染色称为\emph{邻点定位$k$-染色}，若对同一颜色类中的任意两个顶点，其邻域中出现的颜色集合不同。邻点定位色数$\chi_{NL}(G)$是使得$G$存在邻点定位$k$-染色的最小$k$值。图$G$的一个正常$k$-染色称为\emph{定位$k$-染色}，若对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$d(x,S_i)\neq d(y,S_i)$。定位色数$\chi_{L}(G)$是使得$G$存在定位$k$-染色的最小$k$值。由此可得对任意图$G$有$\chi(G)\leq\chi_L(G)\leq\chi_{NL}(G)$，其中$\chi(G)$是$G$的通常色数。我们证明：对任意满足$2\leq p\leq q\leq r$的三个整数$p,q,r$（$2=p=q<r$情况除外），存在连通图$G_{p,q,r}$使得$\chi(G_{p,q,r})=p$，$\chi_L(G_{p,q,r})=q$且$\chi_{NL}(G_{p,q,r})=r$。我们还证明了图$G$的诱导子图的定位色数（相应地，邻点定位色数）可以比$G$本身任意大。Alcon等人证明：当$G$连通且$\chi_{NL}(G)=k$时，$G$的顶点数$n$的上界为$k(2^{k-1}-1)$（该界是紧的）。当$G$的最大度为$\Delta$时，他们还证明了一个更小的$n$的上界，阶数为$k^{\Delta+1}$。我们推广了后者：若$G$的阶为$n$且边数不超过$an+b$，则$n$的上界阶数为$k^{2a+1}+2b$。此外，我们描述了接近该界的此类图的构造方法。