THE LOCATING RAINBOW CONNECTION NUMBERS OF SOME GRAPH CLASSES

• Main Author: Widyastuty Bustan, Ariestha
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/83917
• Institution: Institut Teknologi Bandung

Let k be a positive integer and G = (V (G);E(G)) be a finite and connected graph. A rainbow vertex k-coloring of G is a function c : V (G) ????! f1; 2; :::; kg such that for every u and v in V (G) there exists a rainbow path connecting them. A path P of G whose all internal vertices have distinct colors is called a rainbow vertex path. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest positive integer k so that G has a rainbow vertex k-coloring. For i 2 f1; 2; :::; kg, let Ri be a set of the vertices with color i and = fR1;R2; :::;Rkg be an ordered partition of V (G). The rainbow code of a vertex v 2 V (G) with respect to is defined as the k-tuple rc(v) = (d(v;R1); d(v;R2); :::; d(v;Rk)) where d(v;Ri) = minfd(v; y)jy 2 Rig for i 2 f1; 2; :::; kg. If every vertex of G has a distinct rainbow code, then c is called a locating rainbow k-coloring of G. The locating rainbow connection number of G, denoted by rvcl(G), is defined as the smallest positive integer k such that G has a locating rainbow k-coloring. In this dissertation, strict upper and lower bounds are provided for the locating rainbow connection number of any finite and connected graphs. We also determine the locating rainbow connection numbers of specific graph classes, such as bipartite graphs and transitive vertex graphs. Additionally, the locating rainbow connection numbers of graphs resulting from various graph operations are determined, including amalgamation graphs, generalized barbell graphs, and edge corona graphs.