The Bounds on the Locating-Chromatic Number for a Subdivision of a Graph on One Edge

The study of locating-chromatic numbers has been done for many classes of graphs. Recently, Behtoei and Anbarloei (2014) presented the locating-chromatic number of wheels. Inspired by the result of Behtoei and Anbarloei, the authors (2012,2013) gave the locating-chromatic number of the subdivision of a wheel on one of its spoke or cycle edges. In this paper, we determine an upper bound on the locating-chromatic number of a subdivision of any connected graph on any one edge and show that the bound is tight. In particular, we give the lower bound for the locating-chromatic number of a subdivision of any graph on a pendant edge. Furthermore, we give the exact values of the locating-chromatic number of a subdivision of a complete and a star graph on any one edge.