On the Locating-Chromatic Number of the Sunflower Graph

Abstract

Let c be vertex coloring of a connected graph. Define $c: V \rightarrow {1, 2, ...,k}$ such that $c(u) \neq c(v)$ for adjacent vertices u and $v$ in G. Let S_i be a set of vertices assigned by color i where $1 \leq i \leq k$, defined as color class. Let $Pi ={S_1, S_2, ..., S_k}$ be an ordered partition of V(G) that is induced by colo-ring c, then the representation of vertex v with respect to Pi is called a color code of v, denoted as $c_\Pi(v)$, defined as $c_\Pi(v)=(d(v,S_1),d(v,S_2),\ldots,d(v,S_k))$, where d(v, S_i ) =min{d(v,x) | x \in S_i } for $1 \leq i \leq k$. If all distinct vertices of G have distinct color codes, then c is called a k-locating coloring of G. The locating-chromatic number is defined as the minimum k such that graph G admits a k-locating coloring, denoted by $\chi_L(G)$. In this paper, we determine the locating-chromatic number of the sunflower graph SF_n for $n \geq 3$