Abstract
Let G be a nontrivial, connected, and edge-colored graph of order n ≥ 3, where adjacent edges may be colored the same. Let k be an integer with 2 ≤ k ≤ n. A tree T in G is a rainbow tree if no two edges of T are colored the same. For S ⊆ V G , the Steiner distance d S of S is the minimum size of a tree in G containing S . An edge-coloring of G is called a strong k -rainbow coloring if for every set S of k vertices of G there exists a rainbow tree of size d S in G containing S . The minimum number of colors needed in a strong k -rainbow coloring of G is called the strong k -rainbow index srxk G of G. In this paper, we study the strong 3-rainbow index of edge-amalgamation of graphs. We provide a sharp upper bound for the srx3 of edge-amalgamation of graphs. We also determine the srx3 of edge-amalgamation of some graphs.
Highlights
All graphs considered in this paper are simple, finite, and connected
For S ⊆ V (G), a rainbow S -tree is a rainbow tree that contains the vertices of S
The k -rainbow index rxk(G) of G, introduced by Chartrand et al [3], is the minimum number of colors needed in a k -rainbow coloring of G
Summary
All graphs considered in this paper are simple, finite, and connected. We follow the terminology and notation of Diestel [5]. We determine a sharp upper bound for the strong 3-rainbow index of Edge − Amal(G, e, t). We determine the exact values of the strong 3-rainbow index of Edge − Amal(G, e, t) for some connected graphs G. Given c as a strong 3-rainbow coloring of G and X ⊆ E(G) , let c(X) denote the set of colors assigned to all edges of X. Sharp upper bound for srx3(Edge − Amal(G, e, t) In the following theorem, we provide an upper bound for the strong 3-rainbow index of Edge − Amal(G, e, t). The upper bound in Theorem 2.1 is sharp It can be proven by providing some connected graphs G such that srx3(Edge − Amal(G, e, t)) attains the upper bound. For i ∈ [1, t] , let Ai be a set of edges of path uvi
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