The strong 3-rainbow index of edge-comb product of a path and a connected graph

A tree in an edge-colored connected graph G is a rainbow tree if all of its edges have different colors. Let k be an integer with 2 ≤ k ≤ n and S be a k -subset of V ( G ) . The strong k -rainbow index srx k ( G ) of G is the smallest number of colors required in an edge-coloring of G such that every set S in G is connected by a rainbow tree with minimum size. In this paper, we investigate the srx 3 of edge-comb product of a path and a connected graph, denoted by P on (cid:3) (cid:126)e H . It is obvious that the natural upper bound for srx 3 ( P on (cid:3) (cid:126)e H ) is | E ( P on (cid:3) (cid:126)e H ) | . Hence, we ﬁrst provide graphs H with srx 3 ( P on (cid:3) (cid:126)e H ) = | E ( P on (cid:3) (cid:126)e H ) | , then provide a sharper upper bound for srx 3 ( P on (cid:3) (cid:126)e H ) where srx 3 ( P on (cid:3) (cid:126)e H ) (cid:54) = | E ( P on (cid:3) (cid:126)e H ) | . We also provide the exact values of srx 3 ( P on (cid:3) (cid:126)e H ) for some graphs H .


Introduction
Throughout this paper, all graphs are finite, simple, and connected. The terminology and notation refer to Diestel [11]. For simplifying, we define a set [a, b] = {x : a ≤ x ≤ b}. Let G(V, E) be an edge-colored graph of order n ≥ 3. A tree in G is a rainbow tree if all of its edges have different colors. Let k be an integer with k ∈ [2, n]. The smallest number of colors required in an edge-coloring of G such that every k-subset S of V (G) is connected by a rainbow tree is www.ejgta.org The strong 3-rainbow index of edge-comb product of a path and a connected graph | Z. Y. Awanis et al. also investigated the srx 3 of some certain graphs (see [1,2,3]). The following theorems are needed.
Theorem 1.2. [3] Let L n be a ladder graph of order 2n (n ≥ 3). Then srx 3 (L n ) = n. Theorem 1.3. [3] Let K n,n be a regular complete bipartite graph of order 2n (n ≥ 3). Then srx 3 (K n,n ) = n. Theorem 1.4. [3] Let C n be a cycle of order n ≥ 3. Then   Theorem 1.5. [1] Let F n be a fan of order n + 1 (n ≥ 3). Then srx 3 (F n ) = 3, for n = 4; n 2 , otherwise. The following definition of edge-comb product of two graphs is referred to [5]. Given an undirected graph G, an orientation of G is an assignment of a direction to every edge of G. Let G and H be two connected graphs. Let O be an orientation of G and e be an oriented edge of H. The edge-comb product of G and H on e (under the orientation O), denoted by G o £ e H, is a graph formed by taking one copy of G and |E(G)| copies of H and identifying the i-th copy of H at the edge e to the i-th edge of G, where the two edges have the same orientation.
In this paper, we investigate the strong 3-rainbow index of P o n £ e H. In Section 2, we first provide graphs H with srx 3 (P o n £ e H) = |E(P o n £ e H)|, then we provide a sharper upper bound for srx 3 (P o n £ e H). In Section 3, we determine the exact value of srx 3 (P o n £ e H) for some graphs H. In Section 4, we give concluding remarks and some open problems for further investigation.

Sharp upper bound for srx 3 (P o
n £ e H) For two integers n, m ≥ 3, let P o n be a path P n = v 1 v 2 . . . v n of order n with orientation O, where every edge of P n has an orientation from v i to v i+1 for each i ∈ [1, n − 1], and H be a connected graph of order m with V (H) = {w 1 , w 2 , . . . , w m } and e = w a w b be an oriented edge of H which has an orientation from w a to w b . Now, we consider graphs P o n £ e H. For i ∈ [1, n − 1], let the i-th copy of H is denoted by For further discussion, if c is a strong 3-rainbow coloring of P o n £ e H, then the set of colors assigned to the edges in X is denoted by c(X). By considering any three vertices of P n and using Theorem 1.1, we have |c(E(P n ))| = n − 1. ( According to (1), the natural upper bound for srx 3 The following theorem shows that srx 3 Theorem 2.1. For two integers n, m ≥ 3, let P n and T m be a path of order n and a tree of order m, respectively. Let e be any oriented edge of T m . Then srx 3 The following two cases show that there is an edge of C i g for each i ∈ [1, 2] which is not contained in a v 2 − v t i geodesic for any t ∈ [1, g].
It means v 2 = v 1 i . Thus, we have l p i = p − 1 for p ∈ [1, g 2 + 1] and l p i = g − p + 1 for p ∈ [ g 2 + 2, g]. If g is odd, then v ∈ V (C i g ) for some i ∈ [1, 2] We define several sets as follows.
Note that regardless the parity of g, we have either Subcase 2.1 or 2.2 as follows.
Hence, there are two v 2 − v k i geodesics, one path contains v p i v k i and another path contains v k i v q i . Similar to Case 1 for even g, edge v p i v k i can be chosen to be an edge that is not contained in a v 2 − v t i geodesic for any t ∈ [1, g].
Thus, similar to Case 1, we obtain that v According to Cases 1 and 2, there is an edge e i ∈ E(C i g ) for each i ∈ [1,2] such that e i is not contained in a v 2 − v t i geodesic for any t ∈ [1, g]. Therefore, by assigning the color 1 to the edges e 1  According to Theorem 2.2, graph P o n £ e T m is the only graph whose srx 3 is equal to its size. The following theorem provides a sharper upper bound for srx 3 (P o n £ e H).
Theorem 2.3. For two integers n, m ≥ 3, let P n and H be a path of order n and a connected graph of order m, respectively. Let e be any oriented edge of H. Then is also equal to the upper bound given in Theorem 2.3. Thus, the upper bound is sharp. There are other graphs H such that srx 3 . These results are given in Section 3.
3. The strong 3-rainbow index of P o n £ e H for some connected graphs H Our first two results show that there are two connected graphs H such that srx 3 Theorem 3.1. For two integers n, m ≥ 3, let P n and L m be a path of order n and a ladder of order 2m, respectively. Let e be an oriented edge of L m where e = w 1 w m+1 . Then srx 3 (P o n £ e L m ) = m(n − 1).
Proof. Since srx 3 (L m ) = m by Theorem 1.2, it follows by Theorem 2.3 that srx 3 We first verify two properties as follows.
Let e = uv and f = xy, and assume that d(v i , x) < d(v i , y). Observe that edges e and f should be contained in any rainbow Steiner {u, v, y}-tree, but c(e) = c(f ), a contradiction.
Let e = uv and f = xy, and assume that d(v j , x) < d(v j , y). By considering {u, v, y}, we will obtain a contradiction.
According to (1), the natural lower bound for srx 3 Theorem 3.2. For two integers n, m ≥ 3, let P n and K m,m be a path of order n and a regular complete bipartite graph of order 2m, respectively. Let e be any oriented edge of K m,m . Then . By using Theorems 1.3 and 2.3, we have srx 3 . Now, let c be a strong 3-rainbow coloring of P o n £ e K m,m . We first verify two properties as follows.
, vertex w 1 and edge w 1 w i are called the center vertex and the spoke of F m , respectively. In [1], we obtained the following lemma which will be used to prove Theorem 3.3.
Let F m be a fan of order m + 1 (m ≥ 3) which has a strong 3-rainbow coloring. Then each color is assigned to at most two spokes w 1 w i and w 1 w j where w i w j ∈ E(F m ). Proof. Let c be a strong 3-rainbow coloring of P o n £ e F m . Similar to the proof of Theorem 3.2, we have two properties as follows.
Now, we distinguish two cases.
. Now, observe that identifying vertex v 2 in a rainbow Steiner {v 2 , v 4 1 , v 5 1 }-tree and a rainbow v 2 − v n geodesic will obtain a rainbow Steiner {v 4 1 , v 5 1 , v n }-tree. Similarly, identifying vertex v 2 in a rainbow Steiner {v 2 , v 4 1 , v 5 1 }-tree and a rainbow v 2 − v 5 i geodesic for all i ∈ [2, n − 1] will obtain a rainbow Steiner {v 4 1 , v 5 1 , v 5 i }-tree. Since these rainbow Steiner trees must contain edge v 4 1 . This means we have two colors, which are 1 and n, to be assigned to the three edges in a Steiner tree containing {v 2 , v 4 1 , v 5 1 }, which is impossible. Thus, , then it is not hard to find a rainbow Steiner S-tree. Hence, there are two possible sets S as follows. First, we consider case when two vertices of S belong to the same fan F i 4 for some i ∈ [1, n − 1]. Let y ∈ V (F j 4 ) for j ∈ [1, n − 1] with j = i. For i < j, let P be a v i+1 − v j geodesic. Then there is a rainbow Steiner S-tree as given in Table 1. The proof for i > j is similar to the case for i < j.
Next, we consider case when each vertex of S belongs to three different fans F i 4 , F j 4 , and F k 4 for i, j, k ∈ 1[, n − 1]. Without loss of generality, let i < j < k. 4,5], and P k = v k z. Then the tree T = P ∪ P a i ∪ P b j ∪ P k www.ejgta.org The strong 3-rainbow index of edge-comb product of a path and a connected graph | Z. Y. Awanis et al.
with a, b ∈ [1, 2] is a rainbow Steiner S-tree, where the values of a and b depend on the values of p and q, respectively. Note that the case when S contains at least one vertex of P n has been proven. An illustration of a strong 3-rainbow coloring of P o 5 £ e F 4 is given in Figure 2.  Proof. Without loss of generality, let e = w 1 w 2 such that v 1 i = v i and v 2 i = v i+1 for each i ∈ [1, n − 1]. We consider several cases.

Case 1. m is odd
We distinguish two subcases.

Subcase 1.1. m = 5
Suppose that srx 3 (P o n £ e C 5 ) ≤ 2n + n 2 − 3. Let c : E(P o n £ e C 5 ) → [1, 2n + n 2 − 3] be a strong 3-rainbow coloring of P o n £ e C 5 . Observe that c(v i v 5 i ) / ∈ c(E(P n )) and c(v i v 5 i ) = c(v j v 5 j ) for i, j ∈ [1, n − 1] with i = j. Hence, by using (2), we need at least 2n − 2 different colors assigned to all edges v i v i+1 and v i v 5 i for i ∈ [1, n − 1], implying that we have at most n 2 − 1 colors left. Let X be the set of these n 2 − 1 colors. Now, we consider edges , j = i, j = i + 1, and p ∈ [3,4], we obtain that these three edges can not be assigned with colors 3,4], these three edges also can not be assigned with Since every two adjacent edges in C i 5 must have different colors, this forces and at least one edge of edges i should be assigned with colors from X. This condition implies there are two possible proofs that might happen. Before we proceed further, we consider the following two properties.
. Now, we consider these two possible proofs, which are: . If the first case happens, then observe that there are at most n 2 − 1 pairs of two edges Hence, by using (D1), there are at least n − n 2 pairs of two edges (3). However, by using (D2), we need at least n − n 2 different colors assigned to the edges v 4 j v 5 j for all j ∈ [1, n − 1] with j = i, which is impossible since |X| ≤ n 2 − 1. A similar argument applies if the second case happens. For the upper bound, we first define an edge-coloring c of P o n £ e C 5 using 2n + n 2 − 2 colors as follows.

For each
. Now, let S be a 3-subset of V (P o n £ e C 5 ). Since the edge-coloring c assigns 3 different colors to all edges of C i 5 and has the same coloring pattern as given in Figure 1, there is a rainbow Steiner S-tree if S ⊆ V (C i 5 ) for some i ∈ [1, n − 1]. Hence, we distinguish two cases. www.ejgta.org The strong 3-rainbow index of edge-comb product of a path and a connected graph | Z. Y. Awanis et al.

First, we consider
. Then there is a rainbow Steiner S-tree as given in Table 2. Meanwhile for i > j, let P be a v j+1 − v i geodesic, 3,4], and P 4 [3,4]. Then there is a rainbow Steiner S-tree as given in Table 2. Note that the value of b for each case depends on the value of r ∈ [3,5].
. Without loss of generality, let i < j < k. Let P be a v i+1 − v k geodesic, and q ∈ [4,5], 4,5]. Then the tree T = P ∪ P a i ∪ P b j ∪ P c k with a, c ∈ [1, 2] and b ∈ [1, 4] is a rainbow Steiner S-tree, where the values of a and c depend on the values of p and r, respectively, and the value of b depends on the values of j and q.
Note that the case when S contains at least one vertex of P n has been proven for each of the above cases. Figure 3 illustrates a strong 3-rainbow coloring of P o 5 £ e C 5 .   We first consider the following properties. Let c be a strong 3-rainbow coloring of P o n £ e C m .
and v i v 8 i should be assigned with different colors, which means every color in c(X i ) should be used to color these edges. Next, we consider edges However, there is no rainbow Steiner {v i+1 , v 4 i , v 7 i }-tree, a contradiction. For m ≥ 10, it is clearly that |c(X i )| ≥ m − 3 by Theorem 1.4. Now, we distinguish three subcases.
i v 6 i ) = 3(n−1)+1, and assign the colors 3(n−1)+2, 3(n−1)+3, . . . , 5(n−1)+1 to the remaining 2(n − 1) edges of P o n £ e C 8 . Now, let S be a 3-subset of V (P o n £ e C m ). Similar to the proof of Subcase 1.1, we distinguish two cases. First, we consider 2]. Then there is a rainbow Steiner S-tree as given in Table 3, where the value of b depends on the value of r ∈ [3, m]. The proof for i > j is similar to the case for i < j.
Next, without loss of generality, we consider S = {v p i , v q j , v r k } for i, j, k ∈ [1, n − 1] with i < j < k. Let P be a v i+1 − v k geodesic, Then the tree T = P ∪ P a i ∪ P b j ∪ P c k with a, b, c ∈ [1, 2] is a rainbow Steiner S-tree, where the values of a, b, and c depend on the values of p, q, and r, respectively.
Note that the case when S contains at least one vertex of P n has been proven for each of the above cases. Figure 6 illustrates the strong 3-rainbow colorings of P o 5 £ e C m for m ∈ {6, 8}.

Conclusion
We have shown that H is a tree if and only if srx 3 (P o n £ e H) = |E(P o n £ e H)|. Further, we have also provided a sharper upper bound for srx 3 (P o n £ e H), that is srx 3 (P o n £ e H) ≤ srx 3 (H)(n − 1), and have determined the exact values of srx 3 (P o n £ e H) for some connected graphs H. There are many classes of connected graphs H for which the srx 3 (P o n £ e H) is not known. Hence, it is interesting to continue the study by determining the exact value of srx 3 (P o n £ e H) for other connected graphs H. These results are expected to help characterize the connected graphs H with srx 3 (P o n £ e H) = srx 3 (H)(n − 1). Since a path is one of classes of trees, it is also interesting to study the srx 3 of edge-comb product of a tree and a connected graph.