Neighbor-locating coloring: graph operations and extremal cardinalities

Hernando, Carmen; Mora, Mercè; Pelayo, Ignacio M.; Alcón, Liliana; Gutiérrez, Marisa

doi:10.1016/j.endm.2018.06.023

Cited by 5 publications

(2 citation statements)

References 6 publications

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“…Taking cues from the concepts of location [15], neighbor-location [16] and locally identifying colorings [10], recently, two variants of graph coloring were introduced, namely, locating coloring [8] and neighbor-locating coloring [2,4]. While the former concept has been well-studied since 2002 [4,5,6,7,8,9,13,14,17,18,19]), our focus of study is the latter, which was introduced in 2014 in [4] under the name of adjacency locating coloring, renamed in 2020 in [2] and studied in a few papers since then [1,3,11,12].…”

Section: Introductionmentioning

confidence: 99%

New bounds and constructions for neighbor-locating colorings of graphs

Chakraborty¹,

Foucaud²,

Nandi³

et al. 2023

Preprint

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A proper k-coloring of a graph G is a 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 χ N L (G) is the minimum k for which G admits a neighbor-locating k-coloring. A proper k-coloring of a graph G is a 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 ) = d(y, S i ). The locating chromatic number χ L (G) is the minimum k for which G admits a locating k-coloring. It follows that χ(G) ≤ χ L (G) ≤ χ N L (G) for any graph G, where χ(G) is the usual chromatic number of G.We show that for any three integers p, q, r with 2 ≤ p ≤ q ≤ r (except when 2 = p = q < r), there exists a connected graph G p,q,r with χ(G p,q,r ) = p, χ L (G p,q,r ) = q and χ N L (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 et al. showed that the number n of vertices of G is bounded above by k(2 k−1 − 1), where χ N L (G) = k and G is connected (this bound is tight). When G has maximum degree ∆, they also showed that a smaller upper-bound on n of order k ∆+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.

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Section: Introductionmentioning

confidence: 99%

New bounds and constructions for neighbor-locating colorings of graphs

Chakraborty¹,

Foucaud²,

Nandi³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Taking cues from the above research topics, recently, two variants of graph coloring were introduced, namely, locating coloring [9] and neighbor-locating coloring [2,5]. While the former concept has been well-studied since 2002 [5-10, 19, 20, 23-25]), our focus of study is the latter, which was introduced in 2014 in [5] under the name of adjacency locating coloring, renamed in 2020 in [2] and studied in a few papers since then [1,3,14,17].…”

Section: Introductionmentioning

confidence: 99%

New Bounds and Constructions for Neighbor-Locating Colorings of Graphs

Chakraborty

Foucaud

Nandi

et al. 2023

Lecture Notes in Computer Science

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A proper k-vertex-coloring of a graph G is a 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 χNL(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 locating k-coloring if for each pair of vertices x and y in the same color-class, there exists a color class Si such that d(x, Si) ̸ = d(y, Si). The locating chromatic number χL(G) is the minimum k for which G admits a locating k-coloring. It follows that χ(G) ≤ χL(G) ≤ χNL(G) for any graph G, where χ(G) is the usual chromatic number of G.We show that for any three integers p, q, r with 2 ≤ p ≤ q ≤ r (except when 2 = p = q < r), there exists a connected graph Gp,q,r with χ(Gp,q,r) = p, χL(Gp,q,r) = q and χNL(Gp,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 et al. showed that the number n of vertices of G is bounded above by k(2 k−1 − 1), where χNL(G) = k and G is connected (this bound is tight). When G has maximum degree ∆, they also showed that a smaller upper-bound on n of order k ∆+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.

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