LOCATING CHROMATIC NUMBER OF CIRCULANT GRAPH CN (1,2,...,T)

• Main Author: Maulana, Fatra
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/49766
• Institution: Institut Teknologi Bandung

Let ???? be a ????-coloring of a connected graph ???? and let ?= ????1,????2,…,???????? be a partition of ???? ???? induced by ????. The code of ????????? is the ????-tuple ????? ???? = ???? ????,????1 ,???? ????,????2 ,…,???? ????,???????? where ???? ????,???????? =???????????? ???? ????,???? :????????????? for 1??????????. If every two distinct vertices on ???? have distinct codes, then ???? is called locating-coloring of ????. A locating coloring of ???? with the smallest number of colors used is called minimum locating-coloring and this number is called the locating-chromatic number of ????, denoted by ???????? ???? . In this thesis, we consider the circulant graph ????????(1,2,…,????) of order ?????3 with ?????1. Ghanem et al. [Locating Chromatic Number of Powers of Paths and Cycles, Symmetry 11 (2019), 389] proved that (1) If ?????2????+3,?????1mod ????+1 , then ????????(????????(1,2,…,????))=????+3, (2) If ?????2????+3,?????6,????????? mod ????+1 , and ????? 2, ?????32 , then ????????(????????(1,2,…,????)) ?????+2+????, (3) If ?????2????+3,?????4,????? ????2 mod ????+1 , then ????????(????????(1,2,…,????))?????+ ????+12 . We found counter examples for (1), (2), and (3). We determine the correct lower and upper bound of the locating-chromatic number of circulant graph. We also show that the bounds are sharp. Furthermore, if the order of circulant graph is large enough, we also provide an exact value of ????????(????????(1,2,...,????)).