Graceful Chromatic Number of Unicyclic Graphs

Alfarisi, Ridho; Dafik, Dafik; Prihandini, R. M.; Adawiyah, Robiatul; Albirri, Ermita Rizki; Agustin, Ika Hesti

doi:10.1088/1742-6596/1306/1/012039

Cited by 8 publications

(6 citation statements)

References 3 publications

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“…Beberapa peneliti yang telah melakukan penelitian tentang pewarnaan graceful adalah (English & Zhang, 2017) tentang bilangan kromatik graceful pada graf pohon. Selanjutnya [1] menemukan bilangan kromatik graceful pada keluarga graf unicyclic, yaitu graf tadpole, graf pan, dan graf sun. Selanjutnya [8] tentang bilangan kromatik graceful pada beberapa kelas graf khusus.…”

Section: Pendahuluanunclassified

Pewarnaan Graceful Pada Graf Hasil Operasi Comb Graf Siklus Dan Graf Star

Rudi¹,

Wahab²,

Waluyo³

2023

KDMA

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Section: Pendahuluanunclassified

Pewarnaan Graceful Pada Graf Hasil Operasi Comb Graf Siklus Dan Graf Star

Rudi¹,

Wahab²,

Waluyo³

2023

KDMA

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“…Pelabelan graceful G adalah fungsi injektif f : V (G) → {0, 1, ..., m} yang menghasilkan fungsi bijektif f 0 : E(G) → {1, 2, ..., m} didefinisikan dengan f 0 (uv) = |f (u) − f (v)|. Pewarnaan k-graceful adalah pewarnaan titik dimana f : V (G) → {1, 2, ..., k}, dengan k ≥ 2 yang menginduksi pewarnaan sisi dimana f 0 : E(G) → {1, 2, ..., k − 1} didefinisikan f 0 (uv) = |f (u) − f (v)| [3]. Menurut Mincu dkk (2019) pewarnaan k-graceful dari graf terhubung yang tidak berarah G adalah pewarnaan titik yang menggunakan k warna yang menginduksi pewarnaan sisi, di mana warna untuk sisi (u, v) adalah nilai absolut dari selisih antara warna pada titik yang bersinggungan dengan sisi tersebut [8].…”

Section: Pendahuluanunclassified

“…Penelitian ini telah dilakukan oleh English dan Zhang [5] yang menemukan bilangan kromatik graceful pada graf pohon. Selanjutnya Alfarisi, dkk [3] menemukan bilangan kromatik graceful pada keluarga graf unicyclic (graf tadpole, graf pan, dan graf sun). Selanjutnya Mincu, dkk [8] juga telah menemukan bilangan kromatik graceful pada beberapa kelas graf khusus.…”

Section: Pendahuluanunclassified

Bilangan Kromatik Graceful pada Keluarga Graf Unicyclic

Sania

Dafik

Fatahillah

2020

CGANT. JOU. OF MATH. AND APP.

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Suppose $ G $ is a graph, where $ G = \{V (G), E (G) \} $. Graceful coloring is defined by $ c: V (G) \to \{1,2, ..., k \} $ which induces a proper edge coloring $ c': E (G) \to \{1,2, ..., k- 1 \}$ defined by $c'(xy)=|c(x)-c(y)|$, where $ k \geq 2 $, $ k \in N $. Coloring is said to be graceful if these 3 conditions are satisfied, namely the proper vertex color, the proper edge color, and the edge color, which are the absolute difference between the color of the accident vertex. The subgraph $H$ on that graceful coloring is smaller than the $G$. Furthermore, one of the subgraphs in the unicyclic graph family is a cycle graph. The graceful chromatic number on a graph denoted by $ \chi_g (G) $, is the optimum number of graceful colors from graph $G$. This research aims to find graceful chromatic numbers in the unicyclic graph family, namely bull graphs, net graphs, cricket graphs, caveman graphs, peach graphs, and flowerpot graphs. The results of this study indicate that $\chi_g(C_l) \geq 4$, where $C_l$ is a unicyclic graphs.

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“…Mincu et al in [6] derived graceful chromatic number of some well-known graph classes, such as diamond graph, Petersen graph, Moser spindle graph, Goldner-Harary graph, friendship graphs, and fan graphs. Graceful chromatic number of some particular unicyclic class graphs were presented by Alfarisi et al (2019) in [1].…”

Section: Introductionmentioning

confidence: 99%

Graceful Chromatic Number of Some Cartesian Product Graphs

Suparta,

Venkatachalam,

Gunadi

et al. 2023

UMJ

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A graph $G(V,E)$ is a system consisting of a finite non empty set of vertices $V(G)$ and a set of edges $E(G)$. A (proper) vertex colouring of $G$ is a function $f:V(G)\rightarrow \{1,2,\ldots,k\},$ for some positive integer $k$ such that $f(u)\neq f(v)$ for every edge $uv\in E(G)$. Moreover, if $|f(u)-f(v)|\neq |f(v)-f(w)|$ for every adjacent edges $uv,vw\in E(G)$, then the function $f$ is called graceful colouring for $G$. The minimum number $k$ such that $f$ is a graceful colouring for $G$ is called the graceful chromatic number of $G$. The purpose of this research is to determine graceful chromatic number of Cartesian product graphs $C_m \times P_n$ for integers $m\geq 3$ and $n\geq 2$, and $C_m \times C_n$ for integers $m,n\geq 3$. Here, $C_m$ and $P_m$ are cycle and path with $m$ vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs.

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Graceful Chromatic Number of Unicyclic Graphs

Cited by 8 publications

References 3 publications

Pewarnaan Graceful Pada Graf Hasil Operasi Comb Graf Siklus Dan Graf Star

Pewarnaan Graceful Pada Graf Hasil Operasi Comb Graf Siklus Dan Graf Star

Bilangan Kromatik Graceful pada Keluarga Graf Unicyclic

Graceful Chromatic Number of Some Cartesian Product Graphs

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