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All Journal Science and Technology Indonesia JMPM: Jurnal Matematika dan Pendidikan Matematika BAREKENG: Jurnal Ilmu Matematika dan Terapan Eksakta : Berkala Ilmiah Bidang MIPA Indonesian Journal of Combinatorics M A T H L I N E : Jurnal Matematika dan Pendidikan Matematika Jurnal Matematika UNAND Jurnal Saintika Unpam : Jurnal Sains dan Matematika Unpam Mandalika Mathematics and Educations Journal InPrime: Indonesian Journal Of Pure And Applied Mathematics Warta Pengabdian Andalas: Jurnal Ilmiah Pengembangan Dan Penerapan Ipteks Jurnal Natural Limits: Journal of Mathematics and Its Applications
Des Welyyanti
Departemen Matematika Dan Sains Data, FMIPA, Universitas Andalas
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The Locating Chromatic Number of Zigzag Graph Z_n Vella Sagita Putri; Des Welyyanti; Haripamyu
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 7 No. 2 (2025)
Publisher : Department of Mathematics, Faculty of Sciences and Technology, UIN Syarif Hidayatullah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15408/inprime.v7i2.46443

Abstract

The locating chromatic number is a concept developed from vertex coloring and the partition dimension of a graph, first studied by Chartrand et al. (2002). A connected graph G is said to have a locating coloring when each vertex is assigned a color such that the resulting color code defined by its distances to every color class is unique. The minimum number of colors that satisfies this condition is known as the locating chromatic number, denoted by χ_L (G). This study investigates the value of χ_L for the zigzag graph Z_n with n≥3. Although colorings have been studied for various families of graphs, no explicit characterization of zigzag graphs has been established. Our analysis shows that Z_3 has a locating chromatic number of 3, while for all n≥4, the value increases to 4. These results provide the first complete characterization of locating colorings on zigzag graphs and contribute to the broader study of location-based parameters in graphs with structured topology.Keywords: Locating chromatic number; Zigzag graph; Color code. AbstrakBilangan kromatik lokasi merupakan konsep pengembangan dari pewarnaan titik dan dimensi partisi suatu graf yang pertama kali dikaji oleh Chartrand dkk (2002). Sebuah graf terhubung Gdikatakan memiliki pewarnaan lokasi apabila setiap titik diberi warna sedemikian rupa sehingga kode warna yang dibentuk berdasarkan jaraknya terhadap setiap kelas warna bersifat unik. Banyaknya warna minimum yang memenuhi kondisi tersebut disebut bilangan kromatik lokasi, dilambangkan dengan χ_L (G). Penelitian ini mengkaji nilai χ_L pada graf zig-zag Z_n untuk n≥3. Walaupun sejumlah keluarga graf telah diteliti sebelumnya dalam konteks pewarnaan lokasi, graf zig-zag belum pernah memperoleh karakterisasi yang jelas. Hasil analisis menunjukkan bahwa Z_3 memiliki bilangan kromatik lokasi adalah 3, sedangkan untuk semua n≥4, nilai tersebut menjadi 4. Temuan ini memberikan karakterisasi lengkap pertama untuk pewarnaan lokasi pada graf zig-zag dan memperkaya kajian mengenai parameter lokasi pada graf dengan struktur khusus.Kata Kunci: Bilangan kromatik lokasi; Graf zig-zag; Kode warna. 2020MSC: 05C12, 05C15.
The Locating Chromatic Number of Pentagonal Circular Ladder Graph PCLn Des Welyyanti; Annisa Wahyuni; Haripamyu
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 7 No. 2 (2025)
Publisher : Department of Mathematics, Faculty of Sciences and Technology, UIN Syarif Hidayatullah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15408/inprime.v7i2.46473

Abstract

Locating coloring is a type of vertex coloring applied to connected graphs, where each vertex is assigned a color such that adjacent vertices receive different colors. In this setting, each color corresponds to a color class, which consists of all vertices assigned that color. A central notion in locating coloring is the color code of a vertex, determined by its distances to each color class. A coloring is classified as a locating coloring when every vertex in the graph has a unique color code. The locating chromatic number of a graph is the minimum number of colors needed to achieve such a coloring. The Pentagonal Circular Ladder graph is a structure formed by combining a circular graph with pentagonal components. This article examines the locating chromatic number of the Pentagonal Circular Ladder graph and provides an analysis of the behavior of locating colorings within this graph family.Keywords: Locating chromatic number; Partition; Locating coloring; Color code; Pentagonal Circular Ladder Graph. AbstrakPewarnaan lokasi merupakan jenis pewarnaan titik yang diterapkan pada graf terhubung, di mana setiap titik diberi warna sehingga titik-titik yang bertetangga tidak memiliki warna yang sama. Dalam konteks ini, setiap warna membentuk sebuah kelas warna yang terdiri atas seluruh titik yang diberi warna tersebut. Salah satu konsep utama dalam pewarnaan lokasi adalah kode warna suatu titik, yang ditentukan berdasarkan jaraknya terhadap setiap kelas warna. Suatu pewarnaan disebut pewarnaan lokasi apabila setiap titik dalam graf memiliki kode warna yang berbeda. Bilangan kromatik lokasi dari suatu graf didefinisikan sebagai jumlah minimum warna yang diperlukan untuk menghasilkan pewarnaan semacam ini. Graf Pentagonal Circular Ladder merupakan struktur graf yang dibentuk melalui penggabungan graf lingkaran dengan komponen-komponen pentagonal. Artikel ini mengkaji bilangan kromatik lokasi dari graf Pentagonal Circular Ladder serta memberikan analisis mengenai perilaku pewarnaan lokasi pada keluarga graf tersebut.Kata Kunci: Bilangan kromatik lokasi; Partisi; Pewarnaan lokasi; Kode warna; Graf Pentagonal Circular Ladder. 2020MSC: 05C12, 05C15.
Metric Dimension of Graphs Djembe (Dj_n) Duratul Hayat; Des Welyyanti; Haripamyu Haripamyu
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 8 No. 1 (2026)
Publisher : Department of Mathematics, Faculty of Sciences and Technology, UIN Syarif Hidayatullah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15408/inprime.v8i1.46894

Abstract

Let G=(V,E) be a connected graph. For an ordered set A⊆V(G), the representation of a vertex with respect to A is defined by its distance vector to the vertices of A. A set A is called a resolving set if every pair of distinct vertices in G has distinct representations with respect to A. The minimum cardinality of a resolving set is called the metric dimension of G. In this paper, we determine the metric dimension of the Djembe graph D_(j_n ) for n≥3. By constructing appropriate resolving sets and proving their minimality, we obtain an exact formula for the metric dimension. The obtained value depends on the congruence class of n modulo 4, with a special case occurring when n=7. These results provide a new contribution to the study of metric dimensions for cycle-based graph families and extend the existing literature on graph resolvability. AbstrakMisalkan G=(V,E) adalah suatu graf terhubung. Untuk suatu himpunan terurut A⊆V(G), representasi sebuah simpul terhadap A didefinisikan sebagai vektor jarak simpul tersebut ke setiap simpul dalam A. Himpunan A disebut himpunan pembeda (resolving set) jika setiap pasangan simpul yang berbeda dalam Gmemiliki representasi yang berbeda terhadap A. Kardinalitas minimum dari himpunan pembeda disebut dimensi metrik (metric dimension) dari graf G. Pada artikel ini ditentukan dimensi metrik dari graf Djembe D_(j_n ) untuk n≥3. Metode yang digunakan adalah konstruksi himpunan pembeda dan pembuktian minimalitasnya untuk memperoleh batas atas dan batas bawah yang berimpit. Hasil penelitian menunjukkan bahwa dimensi metrik graf Djembe dapat dinyatakan secara eksak dalam bentuk formula tertutup yang bergantung pada kelas kongruensi n modulo 4, dengan satu kasus khusus ketika n=7. Hasil ini memberikan kontribusi baru dalam kajian dimensi metrik pada keluarga graf berbasis siklus yang belum pernah diteliti sebelumnya serta memperkaya perkembangan teori resolvabilitas graf.Kata Kunci: Dimensi metrik, Himpunan pembeda, Representasi simpul, Graf Djembe, Resolvabilitas graf. 2020MSC: 05C12, 05C76.
Metric Dimension of Maple Leaf Graph Des Welyyanti; Susi Mulyani Putri; Ikhlas Pratama Sandy
EKSAKTA: Berkala Ilmiah Bidang MIPA Vol. 27 No. 01 (2026): Eksakta : Berkala Ilmiah Bidang MIPA (E-ISSN : 2549-7464)
Publisher : Faculty of Mathematics and Natural Sciences (FMIPA), Universitas Negeri Padang, Indonesia

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24036/eksakta/vol27-iss01/622

Abstract

This study determines the metric dimension of the Maple Leaf Graph (Mₚ) for 2 ≤ p ≤ 9 using the concepts of vertex distance and resolving sets. By analyzing the distance representation of each vertex with respect to a resolving set, the minimum resolving set is identified, defining the metric dimension of the graph. Calculations were performed manually to ensure consistency and accuracy.The analysis reveals a tiered linear reduction pattern, where the metric dimension does not increase linearly with p. The main findings are summarized in three theorems: for p = 2 and p = 3, the metric dimension of the Maple Leaf Graph equals p; for p = 4, 5, and 6, it equals p – 1; and for p = 7, 8, and 9, it equals p – 2. These results introduce a new class of graphs and provide theoretical insights into the behavior of metric dimension in multi-cycle constructions, thereby contributing to the development of combinatorial graph theory.
The Locating-Chromatic Number of Disjoint Union of Cycles Des Welyyanti; Muhammad Rafif Fajri; Latifa Azhar Abel; Lyra Yulianti; Aisyah Nurinsani; Dony Permana
Science and Technology Indonesia Vol. 11 No. 3 (2026): July
Publisher : Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.26554/sti.2026.11.3.1046-1053

Abstract

Chartrand et al. introduced the idea of the locating-chromatic number of connected graphs in 2002. Let c be a disconnected graph H with k-coloring. Let S_i be the set of all vertices that get color i and let Phi be the partition of V(H) induced by c. The color code C_Phi(v)=(d(v,S_1), d(v,S_2), ..., d(v,S_k)) of a vertex v, where d(v,S_k)=min{d(v,x)} . The locating k-coloring of H is denoted by c if all vertices in H have unique distinct color codes. Welyyanti et al. in 2014 expanded on this idea so that it also applies to unconnected graphs. In this work, for n=>3 and m=>2, we calculate the locating-chromatic number of the disjoint union of cycles, represented by mC_n.
Co-Authors Abdi Musra Abdurrahman, Fatih Abel, Latifa Azhar Adebli Aulia Zami ADEK HAYATI PUTRI Admi Nazra Afrianti Afrianti Aisyah Khairunnisa Ramadhani Aisyah Nurinsani Aisyah Nurinsani Alifaziz Arsyad Alifaziz Arsyad Angryanof, Uthary Putri Annisa Wahyuni Bunga Citra Mayora Daramenra, Romie Devianto, Dodi Dony Permana Duratul Hayat Efendi Efendi Effendi Effendi Eka Rahayu Nengsih A ELIZIA AUGUSTIN Elva Rahimah Emi Lazola Fadhila Radiah Anas Fajri, Muhammad Rafif Fajri, Muhammad Rafif Fakhri Zikra Fazira Ramadhani Febria Angraini Fitri - Anggalia Hamdi, Muhammad Haripamyu Haripamyu Haripamyu Haripamyu Hazmira Yozza Ikhlas Pratama Sandy Kelson Novrianus Lessya KIKI KHAIRA MARDIMAR Laila Hidayati Laila Hidayati Latifa Azhar Abel Lessya, Kelson Novrianus Mardimar, Kiki Khaira Mega Silvia Meiza Fiqrul Hanif MELATI NUR mellany mellany Mellany, Mellany Muhammad Rafif Fajri Muhammad Rafif Fajri MUHAMMAD RANDA Mulyani Putri, Susi Muthia Muhana Mutiara Ramadhani Syafnur Nada Andriani Nada Andriani Narwen Narwen Noverina Alfiany Nurinsani, Aisyah Nurwijayanti Permana, Dony Raisatun Nuha REFI ALKADAR Rendy Aditya Pratama Romie Daramenra RUVIQA PUTRI SOLEHA Saputro, Suhadi Wido Suci Rahma Putri Sugesti Sugesti Susi Mulyani Putri SUTANTO, DASA Sutra Lidya Pritama Syafruddin . Tika Apriliza Uthary Putri Angryanof Vella Sagita Putri Yanita Yanita Yanuar, Ferra Yulianti, Lyra Zahara Zahara ZULKARNAIN, DEBI