On the Locating Chromatic Number of Certain Barbell Graphs

Asmiati, Asmiati; Yana, I Ketut Sadha Gunce; Yulianti, Lyra

doi:10.1155/2018/5327504

Cited by 10 publications

(7 citation statements)

References 12 publications

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“…Metode pada penelitian ini berupa kajian pustaka khususnya tentang konsep-konsep bilangan kromatik lokasi suatu graf (Chartrand et al, 2002), bilangan kromatik lokasi dari hasil kali Kartesian pada graf (Behtoei & Omoomi, 2016), bilangan kromatik lokasi pada pohon (graf) tertentu yaitu graf 𝑛𝑇 𝑘,𝑚 dan graf kembang api (𝐹 𝑛,𝑘 ) (Asmiati, 2016), bilangan kromatik lokasi dari graf barbel tertentu (Asmiati et al, 2018), serta bilangan kromatik lokasi dari graf bayangan dan graf middle (Kabang et al, 2020). Lebih lanjut, pada bagian pembahasan, ditentukan bilangan kromatik lokasi untuk graf total dari graf bintang 𝑇(𝑆 𝑛 ) dan graf splitting dari graf bintang 𝑆′(𝑆 𝑛 ).…”

Section: Metode Penelitianunclassified

Bilangan Kromatik Lokasi Pada Graf Total Dan Graf Splitting Dari Graf Bintang

Fran

Kabang

Yundari

2021

TEOREMA

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Misal diberikan graf terhubung G = (V, E) dan c menyatakan pewarnaan titik di G sehingga untuk titik u yang bertetangga dengan titik v di G, c(u) ≠ c(v). Jika himpunan titik-titik yang mempunyai warna i untuk I = 1, …, k dinyatakan dengan Ci, maka Ci disebut kelas warna. Lebih lanjut, dapat ditentukan kode warna cπ(u) untuk titik u yaitu k- pasang terurut,cπ(u) = (d(u, C1), d(u, C2), …, d(u, Ck)),dengan d(u, Ci) = min {d(u, x) l x Є Ci} untuk 1 ≤ i ≤ k, k Є N. Jika kode warna masing-masing titik di G berbeda, maka pewarnaan c adalah pewarnaan lokasi. Warna minimum (banyaknya warna) sehingga graf G dapat diwarnai dengan pewarnaan lokasi dinyatakan dengan XL(G), disebut bilangan kromatik lokasi. Pada artikel ini, diperoleh bilangan kromatik lokasi untuk graf total dari graf bintang (dinotasikan T(Sn)) dan graf splitting dari graf bintang (dinotasikan S’(Sn)) yaitu XL(T(Sn)) = n + 2, n = 1,2 dan XL(T(Sn)) = n + 1, n ≥ 3 dan XL(S’(Sn)) = n + 2, n Є N.Kata kunci: Kelas warna, kode warna, pewarnaan lokasi

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Section: Metode Penelitianunclassified

Bilangan Kromatik Lokasi Pada Graf Total Dan Graf Splitting Dari Graf Bintang

Fran

Kabang

Yundari

2021

TEOREMA

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“…However, a number of researches restricted on specific classes of graphs have been carried out. For instance, the locating chromatic number for some families of graphs has been found such as paths, cycles, complete multipartite, and bistars in [15], amalgamation of stars in [3], firecracker graphs in [4], Barbell graphs in [5], Kneser graphs in [12], powers of paths and cycles in [17], book graphs in [19], Origami graphs in [20], and Möobius ladder graphs in [21]. Characterizations of graphs having certain locating chromatic number were studied such as trees with locating chromatic number 3 in [8], trees of order n with locating chromatic number n − t for 2 ≤ t < n 2 in [22], unicyclic graphs of order n with locating chromatic number n − 3 or n − 2 in [2,7], and graphs of order n with locating chromatic number n − 1 in [14].…”

Section: Introductionmentioning

confidence: 99%

The locating chromatic number for m-shadow of a connected graph

Sudarsana

Susanto

Musdalifah

2022

EJGTA

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Let c : V (G) → {1, 2, . . . , k} be a proper k-coloring of a simple connected graph G. Let Π = {C 1 , C 2 , . . . , C k } be a partition of V (G), where C i is the set of vertices of G receiving color i. The color code, c Π (v), of a vertex v with respect to Π is an ordered k-tupleIf distinct vertices have distinct color codes then c is called a locating coloring of G. The minimum k for which c is a locating coloring is the locating chromatic number of G, denoted by χ L (G). Let G be a non trivial connected graph and let m ≥ 2 be an integer. The m-shadow of G, denoted by D m (G), is a graph obtained by taking m copies of G, say G 1 , G 2 , . . . , G m , and each vertex v in G i , i = 1, 2, . . . , m−1, is joined to the neighbors of its corresponding vertex v ′ in G i+1 . In the present paper, we deal with the locating chromatic number for m-shadow of connected graphs. Sharp bounds on the locating chromatic number of D m (G) for any non trivial connected graph G and any integer m ≥ 2 are obtained. Then the values of locating chromatic number for m-shadow of complete multipartite graphs and paths are determined, some of which are considered to be optimal.

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“…Motivated by the result of Asmiati et al [3] about the determination of the locating chromatic number of certain barbell graphs, in this paper we determine the locating chromatic number of shadow path graphs and barbell graph containing shadow path for n ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

On the Locating Chromatic Number of Barbell Shadow Path Graph

Asmiati

Damayanti²,

Yulianti³

2021

IJC

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The locating-chromatic number was introduced by Chartrand in 2002. The locating chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph. The locating chromatic number of a graph is defined as the cardinality of the minimum color classes of the graph. In this paper, we discuss about the locating-chromatic number of shadow path graph and barbell graph containing shadow graph.

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On the Locating Chromatic Number of Certain Barbell Graphs

Cited by 10 publications

References 12 publications

Bilangan Kromatik Lokasi Pada Graf Total Dan Graf Splitting Dari Graf Bintang

Bilangan Kromatik Lokasi Pada Graf Total Dan Graf Splitting Dari Graf Bintang

The locating chromatic number for m-shadow of a connected graph

On the Locating Chromatic Number of Barbell Shadow Path Graph

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