The strong 3-rainbow index of edge-amalgamation of some graphs

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.


Introduction
All graphs considered in this paper are simple, finite, and connected. We follow the terminology and notation of Diestel [5]. For simplifying, we define [a, b] as a set of all integers x with a ≤ x ≤ b. Let G be an edge-colored graph of order n ≥ 3, 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. For S ⊆ V (G) , a rainbow S -tree is a rainbow tree that contains the vertices of S . Let k be an integer with k ∈ [2, n]. An edge-coloring of G is called a k -rainbow coloring if for every set passwords along the line are distinct? This situation can be modeled by a graph and the minimum number of these passwords is represented by the k -rainbow index of a graph.
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 a Steiner S -tree. The maximum Steiner distance of S among all sets S of k vertices of G is called the k -Steiner diameter sdiam k (G) of G . Chartrand et al. [3] stated that for every connected graph G of order n ≥ 3 and each integer k with k ∈ [3, n], k − 1 ≤ sdiam k (G) ≤ rx k (G) ≤ n − 1. In [3], they showed that trees are composed of a class of graphs whose k -rainbow index attains the upper bound for rx k (G) .
They also determined the k -rainbow index of cycles and the 3 -rainbow index of complete graphs. Chen et al. [4] provided the 3 -rainbow index of regular complete bipartite and multipartite graphs and wheels. In [1], we determined the 3 -rainbow index of amalgamation of some graphs with diameter 2. Liu and Hu in 2014 [10] studied the 3 -rainbow index with respect to three important graph product operations, namely the Cartesian product, strong product, and lexicographic product, and also other graph operations. Graph operations are an interesting subject, which can be used to understand structures of graphs.
We generalized the concept of the k -rainbow index of G called the strong k -rainbow index of G * . A strong k -rainbow coloring of G is an edge-coloring of G having the property that for every set S of k vertices of G there exists a rainbow tree of size d(S) containing S . Such a rainbow tree is called a rainbow Steiner S -tree. The minimum number of colors needed in a strong k -rainbow coloring of G is the strong k -rainbow index of G , denoted by srx k (G). Thus, we have rx k (G) ≤ srx k (G) for every connected graph G . If k = 2 , then srx 2 (G) is the strong rainbow connection number src(G) of G [2]. Chartrand et al. [2] gave lower and upper bounds for the strong rainbow connection number; that is, diam(G) ≤ rc(G) ≤ src(G) ≤ |E(G)|.
Note that every coloring that assigns distinct colors to all edges of a connected graph is a strong krainbow coloring. Thus, the strong k -rainbow index is defined for every connected graph G . Furthermore, if G is a nontrivial connected graph of size |E(G)| whose k -Steiner diameter is sdiam k (G), then it is easy to check that We have determined the strong 3-rainbow index of some certain graphs. We also provided a sharp upper bound for the strong 3-rainbow index of amalgamation of graphs and determined the exact values of the strong 3 -rainbow index of amalgamation of some graphs * . The following results are needed.
Theorem 1.2 * For n ≥ 3 , let L n be a ladder graph of order 2n . Then srx 3 (L n ) = sdiam 3 (L n ) = n . Theorem 1.3 * For n ≥ 3 , let K n,n be a regular complete bipartite graph of order 2n . Then srx 3 (K n,n ) = n . Theorem 1.4 * Let C n be a cycle of order n ≥ 3. Then: [4,6] or n = 8; n, for n = 7 or n ≥ 9.
For illustration, strong 3-rainbow colorings of C 3 , C 4 , C 5 , C 6 , and C 8 are given in Figure 1.  In this paper, we study graphs of type Edge − Amal(G, e, t) . It is needed when we want to make a larger and complex communication networks and some agencies must pass through one or two centers in order to transfer information or communicate with each other safely. We focus on k = 3 . We determine a sharp upper bound for the strong 3-rainbow index of Edge − Amal(G, e, t). We also determine the exact values of the strong 3-rainbow index of Edge − Amal(G, e, t) for some connected graphs G .

Main results
Let G be a simple connected graph of order n ≥ 3 and let e be a terminal edge of G , which has an orientation.
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 . For t ≥ 2, consider graphs Edge − Amal(G, e, t) . Let V (Edge − Amal(G, e, t)) = {u, v} ∪ {v p i |i ∈ [1, t], p ∈ [1, n − 2]} and uv be the identified edge of Edge − Amal(G, e, t) . For further discussion, given a tree T of size m as a subgraph of Edge − Amal(G, e, t) , let T = {e 1 , e 2 , ..., e m } denote the tree with edge set {e 1 , e 2 , ..., e m } .

Sharp upper bound for srx
In the following theorem, we provide an upper bound for the strong 3-rainbow index of Edge − Amal(G, e, t) . Theorem 2.1 Let t and n be two integers with t ≥ 2 and n ≥ 3 . Let G be a simple connected graph of order n and e be a terminal edge of G . Then: Proof Following (1.1), we know that |E(Edge − Amal(G, e, t))| = t (|E(G)| − 1) + 1 is the natural upper bound for srx 3 (Edge − Amal(G, e, t)). Now, let c ′ be a strong 3 -rainbow coloring of G . We show that srx 3 (Edge−Amal(G, e, t)) ≤ t (srx 3 (G)) by defining a strong 3 -rainbow coloring c : E(Edge−Amal(G, e, t)) → [1, t (srx 3 (G))] as follows: Observe that the coloring c above maintains the position of colors in G i and assigns distinct colors in E(G i ) and E(G j ) for distinct i and j in [1, t] . Therefore, it is not difficult to find a rainbow Steiner S -tree for every set S of three vertices of Edge − Amal(G, e, t) .  Proof Note that the edge-amalgamation of trees is also a tree with |E(Edge − Amal(T n , e, t)) Let C n be a cycle of order n ≥ 3 . Consider graphs Edge − Amal(C n , e, t) where e is an arbitrary edge of We start with the following observation, which will be used to prove the lower bound in Theorem 2.4.
. If c is a strong 3-rainbow coloring of Edge − Amal(C n , e, t), then: Proof Let i and j be two distinct integers in [1, t].

By considering {v
2 Theorem 2.4 Let t be an integer at least 2 and n be an odd integer at least 9. Let C n be a cycle of order n and e be an arbitrary edge of C n . Then srx 3 (Edge − Amal(C n , e, t)) = t (n − 1) + 1 .
Proof Since t (srx 3 (C n )) = t n by Theorem 1.4 and t (|E(C n )| − 1) + 1 = t (n − 1) + 1 , it follows by Theorem 2.1 that srx 3 (Edge − Amal(C n , e, t)) ≤ t (n − 1) + 1 . Thus, we only need to prove the lower bound. Following For n ≥ 3 , recall that a fan F n of order n + 1 is a graph constructed by joining a vertex v to every vertex of a path P n : v 1 v 2 ...v n . The edges of P n are called the rims of F n and the edges connecting v to the vertices of P n are called the spokes of F n . Before we proceed to Theorem 2.8, we first determine the strong 3 -rainbow index of F n . We start with the following lemma. Proof Suppose that there are three spokes of F n , vv i , vv j , and vv k , such that c(vv i ) = c(vv j ) = c(vv k ) .
Note that two of the three vertices v i , v j , and v k are not adjacent. Without loss of generality, assume that v i , a contradiction. Hence, at most two spokes of F n may be colored the same. Furthermore, if The following theorem is an immediate consequence of Lemma 2.5.

Theorem 2.6
For n ≥ 3 , let F n be a fan of order n + 1 . Then: , which can be obtained by assigning the color 1 to the edges vv 1 , vv 2 , and v 2 v 3 , and the color 2 to the edges vv 3 3], which can be obtained by assigning the color 1 to the edges vv 1 , vv 2 , and v 2 v 3 , the color 2 to the edges vv 3 , vv 4 , and v 1 v 2 , and the color 3 to the edge v 3 v 4 . By these two colorings, it is easy to find a rainbow Steiner S -tree for every set S of three vertices of F n for n ∈ [3,4].
For n ≥ 5, it follows by Lemma 2.5 that srx 3 as follows: Now we show that c is a strong 3-rainbow coloring of F n . Let S be a set of three vertices of F n . Let i , j , and k be three distinct integers in [1, n]. We consider two cases.
Now we consider graphs Edge − Amal(F n , e, t) where e = vv s is an arbitrary spoke of F n . By symmetry, The following observation is also an immediate consequence of Lemma 2.5.

Observation 2.7
Let t , n , and s be three integers with t ≥ 2 , n ≥ 3 , and s ∈ Proof Since t (srx 3 (F n )) = t (⌈ n 2 ⌉) by Theorem 2.6 and t (|E(F n )| − 1) + 1 = t (2n − 2) + 1 , it follows by Theorem 2.1 that srx 3 (Edge − Amal(F n , e, t)) ≤ t (⌈ n 2 ⌉). Thus, we only need to prove the lower bound. Let c be a strong 3-rainbow coloring of Edge − Amal (F n , e, t) G, e, t) for some connected graphs G In this subsection, we determine the strong 3-rainbow index of Edge − Amal (G, e, t) for some connected graphs G . In particular, we consider G as a cycle, a fan, a ladder, and a regular complete bipartite graph. First, we consider graphs Edge − Amal(C n , e, t) where e is an arbitrary edge of C n . In Theorem 2.4, we determine srx 3 (Edge − Amal(C n , e, t)) for odd n ≥ 9 . In the next theorem, we determine srx 3 (Edge − Amal(C n , e, t)) for other values of n . First, we verify the following observation. i . If c is a strong 3 -rainbow coloring of Edge − Amal(C n , e, t) , then: Proof Let i and j be two distinct integers in [1, t]. 2 Theorem 2.10 Let t and n be two integers with t ≥ 2 and n ≥ 3. Let C n be a cycle of order n and e be an arbitrary edge of C n . Then: , for n = 3, or n = 5 and t ≥ 3; t (srx 3 (C n ) − 1) + 1, for n = 4, or n = 5 and t = 2; t (srx 3 (C n ) − 2) + 2, for even n ≥ 6, or n = 7 and t = 2; 5t + 1, for n = 7 and t ≥ 3.
Proof For each i ∈ [1, t] , let C i n denote the ith cycle C n in Edge − Amal(C n , e, t) . For simplyfing the proof,
Then srx 3 (Edge − Amal(C 3 , e, t)) ≥ t by Observation 2.3. Now we show that srx 3 (Edge − Amal(C 3 , e, t)) ≤ t by defining a strong 3 -rainbow coloring c : E(Edge − Amal(C 3 , e, t)) → [1, t] . This coloring can be obtained by assigning the colors i to the edges uv 1 i for all i ∈ [1, t] , the colors i+1 to the edges vv 1 i for all i ∈ [1, t−1], and the color 1 to the edges uv and vv 1 t . Now we show that c is a strong 3-rainbow coloring of Edge − Amal(C 3 , e, t) . Let S be a set of three vertices of Edge − Amal(C 3 , e, t). We can find a rainbow Steiner S -tree as shown in Table 1.

A set of three vertices S Condition
A rainbow Steiner S-tree By using Theorem 1.4, srx 3 (C 4 ) = 2 . Let c be a strong 3 -rainbow coloring of Edge − Amal(C 4 , e, t) .
Since c(uv) ̸ = c(uv 1 i ) for all i ∈ [1, t], it follows by Observation 2.9 that srx 3 (Edge − Amal(C 4 , e, t)) ≥ t + 1 . Next, we show that srx 3 (Edge − Amal(C 4 , e, t)) ≤ t + 1 . We define an edge-coloring c : E(Edge − Amal(C 4 , e, t)) → [1, t + 1] , which can be obtained by assigning the color 1 to edges uv and v 1 i v 2 i for all i ∈ [1, t] and the colors i + 1 to edges uv 1 i and vv 2 i for all i ∈ [1, t]. Now we show that c is a strong 3-rainbow coloring of Edge − Amal(C 4 , e, t) . Let S be a set of three vertices of Edge − Amal(C 4 , e, t) . Observe that the coloring above assigns two colors to C i 4 and has the same pattern as the strong 3 -rainbow coloring of C 4 as shown in Figure 1. It follows by Theorem 1.4 that we can find a rainbow Steiner S -tree if the vertices of S are contained on the same cycle C i 4 for some i ∈ [1, t] . Hence, we may assume that vertices of S are not contained on the same cycle C i 4 . Let i, j , and k be three distinct integers in [1, t]. By symmetry, we consider six subcases as shown in Table 2. A set of three vertices S A rainbow Steiner S-tree Note that srx 3 (C 5 ) = 3 by Theorem 1.4. For t = 2 , since sdiam 3 (Edge − Amal(C 5 , e, 2)) = 5 , we have srx 3 (Edge − Amal(C 5 , e, 2)) ≥ 5 by (1.1). Next, we show that srx 3 (Edge − Amal(C 5 , e, 2)) ≤ 5 by defining a strong 3 -rainbow coloring of Edge − Amal(C 5 , e, 2) as shown in Figure 2.
For t ≥ 3, let c be a strong 3-rainbow coloring of Edge − Amal(C 5 , e, t). It follows by Observation 2.3 that srx 3 (Edge − Amal(C 5 , e, t)) ≥ 2t . Next, we show that srx 3 (Edge − Amal(C 5 , e, t)) ≤ 2t by defining a Now we show that c is a strong 3 -rainbow coloring of Edge − Amal(C 5 , e, t) . Let S be a set of three vertices of Edge − Amal(C 5 , e, t) . We consider three subcases.
-The vertices of S belong to the same cycle C i 5 for some i ∈ [1, t] Since the coloring above assigns three colors to C i 5 and has the same pattern as the strong 3-rainbow coloring of C 5 as shown in Figure 1, it follows by Theorem 1.4 that we can find a rainbow Steiner S -tree.
-Two vertices of S belong to the same cycle C i We can find a rainbow Steiner S -tree as shown in Table 3.
-Each vertex of S belongs to distinct cycles C i 5 , C j 5 , and C k We can find a rainbow Steiner S -tree as shown in Table 4. . Let i and j be two distinct integers in [1, t] . By using Observation 2.9, we need at least 2t distinct colors assigned to the edges uv 1 i and v 1 i v 2 i for all i ∈ [1, t] . This implies we have at most one color The proof is similar to the case p = 1, q = 2, r ∈ [1, 3] Table 4.
left, say color a . Note that the only possible rainbow Steiner

this forces c(uv) = a and c(vv
. Now we show that c is a strong 3 -rainbow coloring of Edge − Amal(C 6 , e, t) . Let S be a set of three vertices of Edge − Amal(C 6 , e, t). Since the coloring above assigns four distinct colors to C i 6 and has the same pattern as the strong 3 -rainbow coloring of C 6 as shown in Figure 1, if the vertices of S belong to the same cycle C i 6 for some i ∈ [1, t], then by using Theorem 1.4, there exists a rainbow Steiner S -tree by coloring c. Therefore, we consider the following subcases.
-Two vertices of S belong to the same cycle C i [1,2] and q ∈ [3,4], then P = v p i v 1 i uvv 4 j v q j is a rainbow Steiner S -tree. If p, q ∈ [3,4], then T = v p i v 4 i vv 4 j v q j ∪ {uv} is a rainbow Steiner S -tree. A similar argument applies for We can find a rainbow Steiner S -tree as shown in Table  5.
We define an edge-coloring c : as follows: , for even p ∈ [s + 1, n − 2]. By the coloring above, it is not difficult to find a rainbow Steiner S -tree for every set S of three vertices of Edge − Amal(F n , e, t) .
Case 2 n = 4 By using an argument similar to that used in the proof of the lower bound for even n ≥ 6 , we have By the coloring above, it is easy to find a rainbow Steiner S -tree for every set S of three vertices of

t). 2
A ladder graph L n is a Cartesian product of a P n and a P 2 , where P n is a path of order n . Let In the following theorem, we determine the strong 3 -rainbow index of Edge − Amal(L n , e, t) where e is an arbitrary edge of L n .

Theorem 2.12
Let t and n be two integers with t ≥ 2 and n ≥ 3. Let L n be a ladder of order 2n and e be an arbitrary edge of L n . Then srx 3 (Edge − Amal(L n , e, t)) = t (n − 1) + 1.
Proof Without loss of generality, we consider two cases.
First, we prove the lower bound. Assume to the contrary that srx 3 (Edge − Amal(L n , e, t)) ≤ t (n − 1).
Then there exists a strong 3 -rainbow coloring c : ) for all i ∈ [1, t]. This forces us to need one new distinct color assigned to the edge uv , which is impossible. Thus, srx 3 (Edge − Amal(L n , e, t)) ≥ t (n − 1) + 1 .
Next, we consider graphs Edge − Amal (K n,n , e, t) where e is an arbitrary edge of K n,n . We determine the strong 3 -rainbow index of Edge − Amal(K n,n , e, t) , which is given in the following theorem. Theorem 2.13 Let t and n be two integers with t ≥ 2 and n ≥ 3. Let K n,n be a regular complete bipartite graph of order 2n and e be an arbitrary edge of K n,n . Then srx 3 (Edge − Amal(K n,n , e, t)) = t (n − 1) + 1.
Proof Let V (Edge − Amal(K n,n , e, t)  By the coloring above, it is not difficult to find a rainbow Steiner S -tree for every set S of three vertices of Edge − Amal(K n,n , e, t) . 2