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All Journal Natural Science: Journal of Science and Technology Electronic Journal of Graph Theory and Applications (EJGTA) JURNAL ILMIAH MATEMATIKA DAN TERAPAN BAREKENG: Jurnal Ilmu Matematika dan Terapan Mathvision : Jurnal Matematika Riemann : Research of Mathematics and Mathematics Education Jurnal Pengabdian Farmasi dan Sains
Selvy Musdalifah
Algebra And Combinatorics Research Group, Department Of Mathematics, Faculty Of Mathematics And Natural Sciences, Universitas Tadulako
Agriculture, Biological Sciences & Forestry Biochemistry, Genetics & Molecular Biology Chemical Engineering, Chemistry & Bioengineering Computer Science & IT Control & Systems Engineering Economics, Econometrics & Finance Education Electrical & Electronics Engineering Energy Engineering Immunology & microbiology Mathematics Mechanical Engineering Medicine & Pharmacology Physics Transportation

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Journal : Electronic Journal of Graph Theory and Applications (EJGTA)

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The locating chromatic number for m-shadow of a connected graph I Wayan Sudarsana; Faisal Susanto; Selvy Musdalifah
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 10, No 2 (2022): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2022.10.2.18

Abstract

Let c : V(G)→{1, 2, …, k} be a proper k-coloring of a simple connected graph G. Let Π = {C1, C2, …, Ck} be a partition of V(G), where Ci is the set of vertices of G receiving color i. The color code, cΠ(v), of a vertex v with respect to Π is an ordered k-tuple (d(v, C2),d(v, C2),…,d(v, Ck)), where d(v, Ci)=min{d(v, x):x ∈ Ci} for i = 1, 2, …, k. If 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 χL(G). Let G be a non trivial connected graph and let m ≥ 2 be an integer. The m-shadow of G, denoted by Dm(G), is a graph obtained by taking m copies of G, say G1, G2, …, Gm, and each vertex v in Gi, i = 1, 2, …, m − 1, is joined to the neighbors of its corresponding vertex v′ in Gi + 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 Dm(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.
Co-Authors Afni, N Agusman Sahari Al Fajri, Iman Andi Hendra Andri Anita Ahmad Kasim Armayani Arsal Desy Lusiyanti Faisal Susanto Fitria Fitria Hartayuni Sain Hervisari, Hervisari Huzaimah Huzaimah I W. Sudarsa I Wayan Sudarsana Iman Al Fajri Iman Al Fajri Ismiyanti, Ismiyanti Jeffry Kusuma Merdekawati, Dian Ayu Nasria Nacong Nikita Nikita, Nikita Noviana Noviana Nurainun Nurainun Nurwijayanti Rizki Amalia S Fatimah, S Sri Nurhayati Tandi, Andri Trisha Magdalena A. Jauvani Wahyuningsi, Sri Werokati, I Yesi, Indah