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- W4318906902 abstract "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 $chi(G)leqchi_L(G)leqchi_{NL}(G)$ 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 $2leq pleq qleq r$ (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." @default.
- W4318906902 created "2023-02-03" @default.
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- W4318906902 date "2023-01-31" @default.
- W4318906902 modified "2023-10-01" @default.
- W4318906902 title "New bounds and constructions for neighbor-locating colorings of graphs" @default.
- W4318906902 doi "https://doi.org/10.48550/arxiv.2301.13557" @default.
- W4318906902 hasPublicationYear "2023" @default.
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