THE (STRONG) 3-RAINBOWINDEX OF AMALGAMATION OF SOME GRAPHS

• Main Author: Yumni Awanis, Zata
• Format: Theses
• Language: Indonesia
• Subjects: Analisis
• Online Access: https://digilib.itb.ac.id/gdl/view/32166
• Institution: Institut Teknologi Bandung

Let G be a nontrivial and connected graph with n vertices, and let k be an integer with 2 k n. Let h 2 N, define an edge h-coloring c : E(G) ! f1; 2; :::; hg where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. A k-rainbow h-coloring of G is an edge h-coloring of G having property that for every set S of k vertices of G, there exists a rainbow tree T such that S V (T). The k-rainbow index of G, denoted by rxk(G), is the minimum h such that G has a k-rainbow h-coloring. The Steiner distance d(S) of a set S of vertices in G is the minimum size of a tree in G containing S. Such a tree is called Steiner tree. The k-Steiner diameter of G, denoted by sdiamk(G), is the maximum Steiner distance of S among all sets of S with k vertices in G. Chartrand et al. have shown that sdiamk(G) rxk(G) n ???? 1. An edge h-coloring of G is called a strong k-rainbow h-coloring if every set S of k vertices of G, there exists a rainbow Steiner tree containing S. The strong k-rainbow index of G, denoted by srxk(G), is the minimum h such that G has a strong k-rainbow h-coloring. For t 2 N with t 2, let fG1;G2; :::;Gtg be a collection of finite, simple, and connected graphs and each Gi has a fixed vertex voi called a terminal. The amalgamation Amal(Gi; voi) is a graph obtained by taking all the G0 is and identifying their terminals. If for all i 2 f1; 2; :::; tg, Gi = G and voi = v, Amal(Gi; voi) denoted by Amal(G; v; t). In this thesis, we give lower and upper bounds for the (strong) 3-rainbow index of Amal(Gi; voi) for any connected graph Gi. Additionally, we determine the (strong) 3-rainbow index of amalgamation of either tree, or ladders, or wheels.