The Locating-Chromatic Number of Origami Graphs

Bilangan kromatik lokasi graf merupakan pengembangan dari konsep dimensi partisi dan pewarnaan titik suatu graf. Banyaknya warna minimum pada pewarnaan lokasi dari graf G disebut bilangan kromatik lokasi graf G. Pada paper ini dibahas tentang bilangan kromatik lokasi graf split lintasan dan graf barbel split lintasan. Metode yang digunakan untuk mendapatkan bilangan kromatik lokasi dari graf tersebut adalah dengan menentukan batas atas dan batas bawahnya. Hasil yang diperoleh bahwa bilangan kromatik lokasi dari graf split lintasan dan barbelnya adalah sama yaitu 4.Kata kunci: bilangan kromatik lokasi, graf split lintasan, graf barbel split lintasan.

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“…Pada graf origami, Irawan dkk. [19] telah berhasil menentukan bilangan kromatik lokasinya dan menganalisis graf barbelnya. Irawan dkk.…”

Section: Pendahuluanunclassified

Bilangan Kromatik Lokasi Graf Split Lintasan

Rahmatalia

Asmiati

Notiragayu

2022

jmi

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“…Next, Welyanti et al investigate the locating chromatic number of a graph comprising two components (Welyyanti et al, 2017). Furthermore, a methodology has been devised to calculate the locating chromatic number for origami graphs 𝑂 𝑚 and their divisions (one point on the outside of the edge) (Irawan et al, 2021). Subsequently (Asmiati et al, 2023) established the locating chromatic numbers for specific operations involving origami graphs.…”

Section: Introductionmentioning

confidence: 99%

The Locating Chromatic Number for the New Operation on Generalized Petersen Graphs N_P(m,1)

Irawan,

Istiani

2024

Sainmatika

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The locating chromatic number is a graph invariant that quantifies the minimum number of colors required for proper vertex coloring, ensuring that any two vertices with the same color have distinct sets of neighbors. This study introduces a new operation on generalized Petersen graphs denoted by N_(P(m,1)), exploring its impact on locating chromatic numbers. Through systematic analysis, we aim to determine the specific conditions under which this operation influences the locating chromatic number and provide insights into the underlying graph-theoretical properties. The method for computing the locating chromatic number for the new operation on generalized Petersen graphs, denoted by N_(P(m,1)), entails determining the lower and upper limits. The results indicate that the locating chromatic number for the new operation on the generalized Petersen graph is 4 for m=4 and 5 for m≥5. The findings contribute to a broader understanding of graph coloring.

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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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The Locating-Chromatic Number of Origami Graphs

Cited by 3 publications

References 14 publications

Bilangan Kromatik Lokasi Graf Split Lintasan

Bilangan Kromatik Lokasi Graf Split Lintasan

The Locating Chromatic Number for the New Operation on Generalized Petersen Graphs N_P(m,1)

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

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