THE LOCATING RAINBOW EDGE CONNECTION NUMBERS OF SOME GRAPH CLASSES

• Main Author: Mukayis
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/87978
• Institution: Institut Teknologi Bandung

Let k be a positive integer and G = (V (G),E(G)) be a finite and connected graph. A rainbow-edge-k-coloring of G is a function c : E(G) ? {1, 2, ..., k} such that for every u and v in V (G) there exists a rainbow edge path that connects them. Let e = uv and f = xy are element of edges in G, denoted by d(e, f), defined as d(e, f) = ( min{d(u, x), d(u, y), d(v, x), d(v, y)} + 1, if e ?= f; 0, if e = f. For i ? {1, 2, ..., k}, let Ri be a set of edge with color i and ? = {R1,R2, ...,Rk} be an ordered partition of E(G). The rainbow code of a edge e ? E(G) with respect to ? is defined as the k?tuple rc?(e) = (d(e,R1), d(e,R2), ..., d(e,Rk)) with d(e,Ri) = min{d(e, y)|y ? Ri} for each i ? {1, 2, ..., k}. If every edge of G has distinct rainbow codes, then c is called a locating rainbow edge k?coloring of G. The locating rainbow edge connection number of G, denoted by recl(G), is defined as the smallest positive integer k such that G has a locating rainbow edge k-coloring, denoted by recl(G). In this thesis, we provide a lower bound and upper bound for the locating rainbow edge connection numbers of a graph. Furthermore, we determine the locating rainbow connection number of some classes graph such as a tree graphs, a cycle graphs, a tadpole graphs, a dumbbell graphs, a triangular snake graphs, and a slanting ladder graphs.