On locating and neighbor-locating colorings of sparse graphs

TL;DR

This paper investigates the properties of neighbor-locating and locating colorings in sparse graphs, establishing upper bounds on graph order based on coloring parameters and proving NP-completeness results for related decision problems.

Contribution

It introduces new bounds on the maximum order of sparse graphs with given neighbor-locating chromatic number and proves NP-completeness for related coloring problems in sparse graphs.

Findings

Upper bounds on graph order based on neighbor-locating chromatic number and average degree.

Construction of graph families approaching these bounds.

NP-completeness of coloring decision problems in sparse graphs.

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 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.