Further Results on Locating Chromatic Number for Amalgamation of Stars Linking by One Path

A. Asmiati (University of Lampung)
Lyra Yulianti (Andalas University)
C. Ike Tri Widyastuti (Fransiskus Senior High School, Bandar Lampung)

Indonesian Journal of Combinatorics
Vol 2, No 1 (2018)

Let G = (V, E) be a connected graph. Let c be a proper coloring using k colors, namely 1, 2, ⋯, k. Let Π = {S1, S2, ⋯, Sk} be a partition of V(G) induced by c and let Si be the color class that receives the color i. The color code, cΠ(v) = (d(v, S1), d(v, S2), ⋯, d(v, Sk)), where d(v, Si) = min{d(v, x)∣x ∈ Si} for i ∈ [1, k]. If all vertices in V(G) have different color codes, then c is called as the locating-chromatic k-coloring of G. Minimum k such that G has the locating-chromatic k-coloring is called the locating-chromatic number, denoted by χL(G). In this paper, we discuss the locating-chromatic number for n certain amalgamation of stars linking a path, denoted by nSk, m, for n ≥ 1, m ≥ 2, k ≥ 3, and k > m.