The Locating-Chromatic Number of Disjoint Union of Cycles

Des Welyyanti (Department of Mathematics and Data Sciences, Faculty of Mathematics and Natural Sciences, Universitas Andalas, Padang, West Sumatra, 25163, Indonesia)
Muhammad Rafif Fajri
Latifa Azhar Abel (Department of Mathematics and Data Sciences, Faculty of Mathematics and Natural Sciences, Universitas Andalas, Padang, West Sumatra, 25163, Indonesia)
Lyra Yulianti (Department of Mathematics and Data Sciences, Faculty of Mathematics and Natural Sciences, Universitas Andalas, Padang, West Sumatra, 25163, Indonesia)
Aisyah Nurinsani (Department of Mathematics and Data Sciences, Faculty of Mathematics and Natural Sciences, Universitas Andalas, Padang, West Sumatra, 25163, Indonesia)
Dony Permana (Department of Statistics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Padang, Padang, West Sumatra, 25171, Indonesia)

Science and Technology Indonesia
Vol. 11 No. 3 (2026): July

Chartrand et al. introduced the idea of the locating-chromatic number of connected graphs in 2002. Let c be a disconnected graph H with k-coloring. Let S_i be the set of all vertices that get color i and let Phi be the partition of V(H) induced by c. The color code C_Phi(v)=(d(v,S_1), d(v,S_2), ..., d(v,S_k)) of a vertex v, where d(v,S_k)=min{d(v,x)} . The locating k-coloring of H is denoted by c if all vertices in H have unique distinct color codes. Welyyanti et al. in 2014 expanded on this idea so that it also applies to unconnected graphs. In this work, for n=>3 and m=>2, we calculate the locating-chromatic number of the disjoint union of cycles, represented by mC_n.