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arXiv · 1806.11465

Neighbor-Locating Colorings in Graphs

Liliana Alcon·Marisa Gutierrez·Carmen Hernando·Merce Mora·Ignacio M. Pelayo
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Abstract

A $k$-coloring of a graph $G$ is a $k$-partition $\Pi=\{S_1,\ldots,S_k\}$ of $V(G)$ into independent sets, called \emph{colors}. A $k$-coloring is called \emph{neighbor-locating} if for every pair of vertices $u,v$ belonging to the same color $S_i$, the set of colors of the neighborhood of $u$ is different from the set of colors of the neighborhood of $v$. The neighbor-locating chromatic number $\chi _{_{NL}}(G)$ is the minimum cardinality of a neighbor-locating coloring of $G$. We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order $n\geq 5$ with neighbor-locating chromatic number $n$ or $n-1$. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.

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Liliana Alcon, Marisa Gutierrez, Carmen Hernando, Merce Mora, Ignacio M. Pelayo. 2018-06-29. Neighbor-Locating Colorings in Graphs. https://arxiv.org/abs/1806.11465

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