The locating chromatic number for m-shadow of a connected graph
| Main Authors: | Sudarsana, I Wayan; Algebra and Combinatorics Research Group (ACRG), Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Tadulako, Indonesia, Susanto, Faisal; Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia, Musdalifah, Selvy; Algebra and Combinatorics Research Group, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Tadulako |
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| Format: | Article info application/pdf eJournal |
| Bahasa: | eng |
| Terbitan: |
GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB
, 2022
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| Subjects: | |
| Online Access: |
https://www.ejgta.org/index.php/ejgta/article/view/1583 https://www.ejgta.org/index.php/ejgta/article/view/1583/pdf_242 |
Daftar Isi:
- 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.