The (Strong) Rainbow Connection Number of Join of Ladder and Trivial Graph

ABSTRACT


A. INTRODUCTION
In the context of graphs, the idea of "rainbow connection" was initially proposed by Chartrand et al. (2008). Let = ( , ) be a graph with vertex set and edge set . A coloring : → {1, 2, … , }, ∈ ℕ, thus, the adjacent edges can share an identical color. Let and be in . An − path in is determined as a rainbow path if each path edge has a different color. Meanwhile, is determined as a rainbow connected if every two vertices and has a rainbow path. Rainbow coloring is the term used to describe an edge color on that connects to the rainbow. If the function employs colors, it constitutes the rainbow −coloring. The smallest positive integer , designated by (G), iis ithe irainbow iconnection inumber iof iG.
Thus, ihas ia irainbow -coloring. Meanwhile, the − rainbow path is considered as the − y rainbow geodesic if the length of the path constitutes the distance ibetween and . becomes a strong rainbow connected if every two vertices and have an -rainbow geodesic. Strong rainbow coloring is iedge icoloring ion i ithat gives a strong rainbow connection. If the function uses colors, it is said to have a strong rainbow −coloring. The lowest positive integer iis ithe istrong irainbow iconnection inumber iof , indicated by ( ). As a result, has a strong rainbow -coloring, and ( ) ≤ i" ( ) ifor iany ico nnected igraph i . On condition that is a rainbow connection, the least diam( ) colors are necessary; the diam( ) refers to the 's diameter. On the other hand, rainbow coloring is defined by if each of its edges is colored differently. Hence, the formula is as follows.
(1) Several previous studies have investigated both the rainbow connection number and strong rainbow connections number. Chartrand et al. (2008) have determined some ( ) and ( ) of connected graphs, as follows. Proposition 1. Let be a nontrivial connected graphs of size . Then 1.
The strong rainbow connection number of stellar graphs, which is ia icorona iproduct iof ia itrivial igraph iand ian i -i copies iladder igraph, was discovered by Shulhany and Salman back in 2016 (Shulhany & Salman, 2016). On the other hand, Fitrianda et al. (2018) have determined a generalized triangular ladder graph's rainbow connection number i and istrong irainbow iconnection inumber. Meanwhile, (L. Chen et al., 2018) present some results of the six rainbow connection parameters. Other previous studies have found other results of (strong) rainbow connection of graph (H. Li et al., 2011;Schiermeyer, 2011).
The concept of rainbow connectivity can apply to data security. Confidential information should be protected from being transferred from one party to another. The security system must be able to prohibit not just unauthorized users from accessing the system, but also users who are already signed in from performing actions that they are not permitted to perform (Morris & Thompson, 1979). A security cracking method known as a rainbow table employs a precalculated table of inverted password hashes to decipher database passwords. The user is verified whether the values match. The rainbow table database is utilized to decrypt the password hash and get authentication (Zhang, Tan & Yu, 2013). To minimize data leakage from such confidential information, each agent should have a different password when transferring information. As a result, a lot of passwords are needed. Fortunately, the rainbow connection concept requires a minimum number of passwords so that two agents can exchange information with different passwords.
Apart from this motivation, the concept of rainbow connection is interesting to study. Many researchers have developed the concept of rainbow connectivity by applying it to various types of graphs. One of them is by performing operations on several graphs and specifying ( ) and ( ) on those graphs. Li and Sun (2012) have obtained several results of the rainbow connection number of graph products, comprising lexicographic products, join, cartesian products, etc. Meanwhile, Resty and Salman (2015) have idetermined ithe irainb ow iconnection inumber iof ithe corona product iof ithe n-crossed prism igraph with ithe itrivial igraph. Septyanto and Sugeng (2017) give lower and upper limits for irainbow iconnectison inumbers iand istrong irainbow iconnection inumbers iby joining two graphs based on individual graph parameters: the number of vertices of degree 0, maximum degree, independent dominance number, clique number, and independent number. Li and Ma (2017) have done related study that icalculated ithe irainbow iconnection inumber iof operating graphs by using several ways such as a) includes a union of graphs, b) adds and removes edges, and c) adds vertices. Strong and accurate values of rainbow connection numbers, as well as their upper bounds of comb product, path, triangular book, circular, or fan graphs are discussed by Dafik et al. (2018). In addition, several previous studies focus on examining rainbow connection numbers from operating graphs (Basavaraju et al., 2014;X. Chen et al., 2019;Fitriani & Salman, 2016;Gembong & Agustin, 2017;Gologranc et al., 2014;Liu, 2014;Maulani et al., 2019;Doan, Ha, and Schiermeyer, 2022).
To two agents to share information while using different passwords, the rainbow connection idea allows for the requirement of a minimum number of passwords. The concept of the rainbow link is intriguing to examine even without this reason. Research on rainbow connection numbers from the join operation of ladder and trivial graphs has never been done. (Kartika, 2020) assert that the ladder graph, denoted by , is obtained from two duplicates of the path , becoming 1 and 2 . The vertex of on 1 is connected to the vertex of on 2 by an edge, with = 1, 2, … , . The trivial graph refers to a graph with only one vertex, denoted

B. METHODS
This research is a literature study. The purpose of this study is to determine the rainbow connection number and strong rainbow connection number from graphs resulting from the join operations of ladder and trivial graphs. The steps in this research are as follows: 1. Defining the problem to be discussed 2. Doing a literature study on rainbow connected numbers and strong rainbow connected numbers 3. Describing a graph resulting from the join operation of a ladder graph and a trivial graph, denoted by ∨ 1 . The vertex set and edge set of ∨ 1 are defined as follows. Definition 1. Let ≥ 3 be an integer. The graph resulting from the join operation of a ladder graph and a trivial graph is a graph with As an illustration, the graph ∨ 1 as shown in Figure 1. Based on the picture above, it can be seen that diam( ∨ 1 ) = 2. 4. Finding a rainbow coloring pattern for ∨ 1 and determining the accurate values of ( ∨ 1 ) and ( ∨ 1 ). 5. Proving the rainbow connection number and the strong rainbow connection number of ∨ 1 . If ( ∨ 1 ) = , ( ∨ 1 ) ≥ and ( ∨ 1 ) ≤ should be done. To prove lower bound of ( ∨ 1 ) ≥ , it is necessary to show a reason for the absence of rainbow coloring with − 1 color or less. Proving upper bound of ( ∨ 1 ) ≤ by constructing a rainbow coloring in ∨ 1 using colors. The same thing is also done for the ( ∨ 1 ). 6. Formulating conclusions based on the results of the theorem analysis that has been proven.

C. RESULTS AND DISCUSSION
In this section, i we calculate ithe irainb ow iconnection inumber iand ithe i strong rainbow iconnection inumber iof the following: the joined ladder graph and trivial graph; according to Theorem 1 and Theorem 2, respectively.

The Rainbow Connection Number of
∨ Theorem 1. Let be a positive integer with an ≥ 3. iThe irainbow iconnection inumber iof ∨ 1 is as follows.
, , −1 , , , The rainbow of 2-coloring of 4 ∨ 1 and the rainbow of 3-coloring of 6 ∨ 1 are illustrated in Figure 2.  Table 1, the rainbow path that connect them is 1 , , 4 . Additionally, suppose 4 and 3 are selected. In the same way, the rainbow path that connects these points is 4 , 4 , 3 . Meanwhile, Figure 2(b) shows the rainbow 3-coloring on graph 6 ∨ 1 . It has been explained in the proof of Theorem 1 that for ≥ 5 it takes 3 colors to color the vertices on graph 6 ∨ 1 so that every two vertices have a rainbow path. For example, vertex 1 and 5 are selected. Based on Table 2, the rainbow path is 1 , , 4 , 5 .
As explained earlier, this rainbow coloring can be applied to security issues. If it is associated with the join operation of a ladder graph with a trivial graph in this study, it can be likened to the point being the center of information. Meanwhile, points and are information agents. To be able to exchange information between one agent and another, you must first pass through the information center. Of course, the concept of strong rainbow coloring provides benefits so that the transfer of information is safer.

D. CONCLUSION AND SUGGESTIONS
This study has determined the exact value of (strong) rainbow connection number of joined ladder graph and trivial graph using the following formulas: ( ∨ 1 ) = ( ∨ 1 ) = 2, for 3 ≤ ≤ 4 and ( ∨ 1 ) = 3, for ≥ 5. This study has discovered ( ∨ 1 ) = ⌈ 2 ⌉ , for ≥ 5. Data security can benefit from the rainbow connectivity concept. Transferring confidential information from one party to another should be prevented. Each agent should use a distinct password when sharing information to reduce data leaking. With the rainbow connection concept, two agents can share information while using separate passwords because it enables the requirement of a minimum number of passwords.