Characterizing all graphs containing cycles with locating-chromatic number 3

Asmiati, Asmiati; Baskoro, Edy Tri

doi:10.1063/1.4724167

Cited by 23 publications

(16 citation statements)

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“…Thus, the function c : V(C m n ) −→ {1, 2, ..., m + t + 1} given by c(x i ) = j for any x i ∈ R j is a locating coloring of C m n . (5) If n = q(m + 1) + (t + i), where 1 ≤ i ≤ t − 2, define R 1 , R 2 , ..., R m+t+1 similar to the case n = q(m + 1) + t except R t+j and R m+3+j , where 1 ≤ j ≤ i. Set R t+j = {x (t+j)+(m+1) , x (t+j)+2(m+1) , ..., x q(m+1)+(t+j) } and R m+j+3 = {x t+j , x (t+j)+q(m+1) }.…”

Section: Proofmentioning

confidence: 99%

“…Hence, the locating chromatic number of the complete graph K n is n. In addition, for paths and cycles of order n ≥ 3, they proved that χ L (P n ) = 3, χ L (C n ) = 3 when n is odd, and χ L (C n ) = 4 when n is even. The locating chromatic numbers of trees, and the amalgamation of stars, the graphs with dominant vertices are studied in [4][5][6], respectively.…”

Section: Introductionmentioning

confidence: 99%

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Locating Chromatic Number of Powers of Paths and Cycles

2019

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Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

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Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Locating Chromatic Number of Powers of Paths and Cycles

2019

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“…Sebagai ilustrasi, diberikan pewarnaan lokasi nS 3,4 untuk 1 ≤ n ≤ 2 yang dapat dilihat pada gambar berikut: Gambar 1. Pewarnaan lokasi minimum pada graf 2S 3,4 Sebagai ilustrasi, diberikan pewarnaan lokasi nS 3,4 untuk n > 2 yang dapat dilihat pada gambar berikut: …”

Section: Buktiunclassified

Bilangan Kromatik Lokasi n Amalgamasi Bintang yang Dihubungkan suatu Lintasan

Asmiati

2017

JMI

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AbstrakBilangan kromatik lokasi graf diperkenalkan oleh Chartrand et al. pada tahun 2002. Konsep ini merupkan perpaduan dari konsep dimensi partisi dan pewarnaan graf.Penentuan bilangan kromatik lokasi suatu graf dilakukan dengan mengkonstruksi batas atasnya dan penentuan batas bawahnya. Dalam paper ini, kami mengkaji tentang bilangan kromatik lokasi amalgamasi bintang tertentu, yang dinotasikanKata kunci: amalgamasi bintang, bilangan kromatik lokasi graf, pewarnaan graf. AbstractThe locating chromatic number of a graph was introduced by Chartrand et al in 2002. This concept is combined from the graph partition dimension and graph coloring. Determining the locating chromatic number of graph with construct the upper bound and the lower bound of the graph. In this paper, we determine the locating chromatic number for certain amalgamation of stars, denoted by nS k,m . We prove that the locating chromatic number, nS k,m for k, m ≥ 2, k ≤ m natural numbers n, k, m is (m + 1), if 1 ≤ n ≤ m k−1 and (m + 2) for n ≤ m k−1 .

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“…Lemma 2.1. [2] Let G be a graph with χ L (G) = k. Then, there are at most k dominant vertices in G and all of them must receive different colors.…”

Section: Basic Propertiesmentioning

confidence: 99%

“…We know that P n , n ≥ 3, is an example of a graph with locatingchromatic number 3. Recently, we characterized all graphs containing a cycle with the locatingchromatic number 3 [2]. In this paper, we will determine all trees with locating-chromatic number 3.…”

Section: Introductionmentioning

confidence: 99%

Characterizing all trees with locating-chromatic number 3

Baskoro

Asmiati

2013

ejgta

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Let c be a proper k-coloring of a connected graph G. Let Π = {S 1 , S 2 , . . . , S k } be the induced partition of V (G) by c, where S i is the partition class having all vertices with color i. The colorIf all vertices of G have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number of G, denoted by χ L (G), is the smallest k such that G posses a locating k-coloring. Clearly, any graph of order n ≥ 2 has locating-chromatic number k, where 2 ≤ k ≤ n. Characterizing all graphs with a certain locating-chromatic number is a difficult problem. Up to now, all graphs of order n with locating chromatic number 2, n − 1, or n have been characterized. In this paper, we characterize all trees whose locating-chromatic number is 3. We also give a family of trees with locating-chromatic number 4.

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Characterizing all graphs containing cycles with locating-chromatic number 3

Cited by 23 publications

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Locating Chromatic Number of Powers of Paths and Cycles

Locating Chromatic Number of Powers of Paths and Cycles

Bilangan Kromatik Lokasi n Amalgamasi Bintang yang Dihubungkan suatu Lintasan

Characterizing all trees with locating-chromatic number 3

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