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On locating and neighbor-locating colorings of sparse graphs

Dipayan Chakraborty, Florent Foucaud, Soumen Nandi, Sagnik Sen, D K Supraja

TL;DR

This paper studies neighbor-locating coloring as a sparse-graph identification problem, establishing a polynomial upper bound on the graph order in terms of the NL-number $k$ and average degree $d$, and providing a near-tight graph construction to illustrate the bound's tightness. It also shows that both locating and neighbor-locating coloring decision problems remain NP-complete for sparse graphs, even under strict sparsity and planarity constraints. The authors clarify the relationships among the ordinary chromatic number, the locating chromatic number, and the neighbor-locating chromatic number, and demonstrate that induced subgraphs can exhibit substantially larger NL or locating numbers. The work extends prior bounds based on maximum degree or cycle rank and opens several avenues for future exploration, including tighter planar bounds and the behavior of these colorings under graph modifications.

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} $χ_{NL}(G)$ 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 $χ_{L}(G)$ 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 $χ_L(G)\le k$ and $χ_{NL}(G)\le k$ 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.

On locating and neighbor-locating colorings of sparse graphs

TL;DR

This paper studies neighbor-locating coloring as a sparse-graph identification problem, establishing a polynomial upper bound on the graph order in terms of the NL-number and average degree , and providing a near-tight graph construction to illustrate the bound's tightness. It also shows that both locating and neighbor-locating coloring decision problems remain NP-complete for sparse graphs, even under strict sparsity and planarity constraints. The authors clarify the relationships among the ordinary chromatic number, the locating chromatic number, and the neighbor-locating chromatic number, and demonstrate that induced subgraphs can exhibit substantially larger NL or locating numbers. The work extends prior bounds based on maximum degree or cycle rank and opens several avenues for future exploration, including tighter planar bounds and the behavior of these colorings under graph modifications.

Abstract

A proper -coloring of a graph is a \emph{neighbor-locating -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} is the minimum for which admits a neighbor-locating -coloring. A proper -vertex-coloring of a graph is a \emph{locating -coloring} if for each pair of vertices and in the same color-class, there exists a color class such that . The locating chromatic number is the minimum for which admits a locating -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 has order , neighbor-locating chromatic number and average degree at most , where is a positive integer, then is upper-bounded by . 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 and 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 -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.
Paper Structure (11 sections, 19 theorems, 26 equations, 4 figures)

This paper contains 11 sections, 19 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Identification problems.
  3. Notation and terminology.
  4. Applications.
  5. Context and contributions.
  6. Organization of the paper.
  7. Gaps among $\chi(G)$, $\chi_L(G)$ and $\chi_{NL}(G)$
  8. Bounds for sparse graphs
  9. Tightness of the obtained bound: proof of Theorem \ref{['graph construction']}
  10. Complexity of Locating coloring and Neighbor-locating coloring for sparse graphs
  11. Conclusions

Key Result

Theorem 1

For all $2 \leq p \leq q \leq r$, except when $p=q=2$ and $r >2$, there exists a connected graph $G_{p,q,r}$ satisfying $\chi(G_{p,q,r}) = p$, $\chi_{L}(G_{p,q,r}) = q$, and $\chi_{NL}(G_{p,q,r}) = r$.

Figures (4)

  • Figure 1: Construction of $G_2$ from $G_1=P_{24}$. Here $\chi_{NL}(P_{24})=4$, the red and blue edges are the two sets of newly added matchings.
  • Figure 2: Construction of $G"_{i+1}$ from $G_i$.
  • Figure 3: The gadget $X_i$ from the construction of $G^*$ from $G$ and its connections.
  • Figure 4: A schematic diagram for the construction of $G^*$.

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • proof
  • Corollary 7
  • proof
  • Theorem 8: alcon2019neighborBA2014
  • ...and 21 more