Abstract
The locating-chromatic number of a graph combines two graph concepts, namely coloring vertices and partition dimension of a graph. The locating-chromatic number is the smallest k such that G has a locating k-coloring, denoted by χL(G). This article proposes a procedure for obtaining a locating-chromatic number for an origami graph and its subdivision (one vertex on an outer edge) through two theorems with proofs.
Highlights
The study of the partition dimension of connected graphs was introduced by Chartrand et al [1,2] with the aim of finding a new method for attacking the problem of determining the metric dimension in graphs
The locating-chromatic number has been determined for a few graph classes; for example, if Pn is a path of order n ≥ 3 the locating-chromatic number is 3; for a cycle Cn if n ≥ 3 is odd, χL(Cn) = 3 was obtained, and if n is even, χL(Cn) = 4 was obtained; for a double star graph (Sa,b), 1 ≤ a ≤ b and b ≥ 2, χL(Sa,b) = b + 1 was obtained
In this article we propose a procedure for obtaining the locating-chromatic number for an origami graph and its subdivision
Summary
The study of the partition dimension of connected graphs was introduced by Chartrand et al [1,2] with the aim of finding a new method for attacking the problem of determining the metric dimension in graphs. The concept of the locating-chromatic number was first introduced by Chartrand et al in 2002, with two obtained graph concepts, namely coloring vertices and partition dimensions of a graph [7]. Irawan and Asmiati in 2018 determined the locating-chromatic number of subdivision firecrackers graphs [20] and in [21] obtained the certain operation of generalized Petersen graphs sP(n, 1).
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