Bilangan Terhubung Titik Pelangi pada Graf Hasil Operasi Korona Graf Prisma (P_(m,2)) dan Graf Lintasan (P_3)

Rainbow vertex-connection number is the minimum k-coloring on the vertex graph G and is denoted by rvc(G). Besides, the rainbow-vertex connection number can be applied to some special graphs, such as prism graph and path graph. Graph operation is a method used to create a new graph by combining two graphs. Therefore, this research uses corona product operation to form rainbow-vertex connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2). The results of this study obtain that the theorem of rainbow vertex-connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2) for 3 = m = 7 are rvc (G) = 2m rvc (G) = 2

Developing A Secure Cryptosystem with Rainbow Vertex Antimagic Coloring of Cycle Graph

An edge labeling of graph G is a function g from the edge set of graph G to the first natural numbers up to the number of the edge set. Graph G admits a rainbow vertex antimagic coloring if, for any two vertices, there is a path with different colors of all internal vertices. The vertex color of graph G is assigned by vertex weight. The vertex weight of graph G is obtained by summing all edge labels that incident with that vertex. The rainbow vertex antimagic connection number of graph G, denoted by rvac(G) is the smallest number of different colors induced by rainbow vertex antimagic coloring. In this research, we determine the upper bound of the rainbow vertex antimagic connection number (rvac)  on a cycle graph (Cn) and create a secured cryptosystem using a modified Affine Cipher based on rainbow vertex antimagic coloring

Total Rainbow Connection Number Of Shackle Product Of Antiprism Graph (〖AP〗_3)

Function if  is said to be k total rainbows in , for each pair of vertex  there is a path called  with each edge and each vertex on the path will have a different color. The total connection number is denoted by trc  defined as the minimum number of colors needed to make graph  to be total rainbow connected. Total rainbow connection numbers can also be applied to graphs that are the result of operations. The denoted shackle graph  is a graph resulting from the denoted graph  where t is number of copies of G. This research discusses rainbow connection numbers rc and total rainbow connection trc(G) using the shackle operation, where  is the antiprism graph . Based on this research, rainbow connection numbers rc shack , and total rainbow connection trc shack for .Fungsi jika c : G → {1,2,. . . , k} dikatakan k total pelangi pada G, untuk setiap pasang titik  terdapat lintasan disebut x-y dengan setiap sisi dan setiap titik pada lintasan akan memiliki warna berbeda. Bilangan terhubung total pelangi dilambangkan dengan trc(G), didefinisikan sebagai jumlah minimum warna yang diperlukan untuk membuat graf G menjadi terhubung-total pelangi. Bilangan terhubung total pelangi juga dapat diterapkan pada graf yang merupakan hasil operasi. Graf shackle yang dilambangkan (G1,G2,…,Gt) adalah graf yang dihasilkan dari graf G yang dilambangkan (G,t) dengan t adalah jumlah salinan dari  Penelitian ini membahas mengenai bilangan terhubung pelangi rc dan bilangan terhubung total pelangi trc(G)menggunakan operasi shackle, dimana G adalah graf Antiprisma (AP3)Berdasarkan penelitian ini, diperoleh bilangan terhubung pelangi rc(shack AP3,t )= t+2, dan total pelangi trc(shack AP3,t)=2t+3 untuk t ≥2

On the Locating Rainbow Connection Number of Trees and Regular Bipartite Graphs

Locating the rainbow connection number of graphs is a new mathematical concept that combines the concepts of the rainbow vertex coloring and the partition dimension. In this research, we determine the lower and upper bounds of the locating rainbow connection number of a graph and provide the characterization of graphs with the locating rainbow connection number equal to its upper and lower bounds to restrict the upper and lower bounds of the locating rainbow connection number of a graph. We also found the locating rainbow connection number of trees and regular bipartite graphs. The method used in this study is a deductive method that begins with a literature study related to relevant previous research concepts and results, making hypotheses, conducting proofs, and drawing conclusions. This research concludes that only path graphs with orders 2, 3, 4, and complete graphs have a locating rainbow connection number equal to 2 and the order of graph G, respectively. We also showed that the locating rainbow connection number of bipartite regular graphs is in the range of r-⌊n/4⌋+2 to n/2+1, and the locating rainbow connection number of a tree is determined based on the maximum number of pendants or the maximum number of internal vertices. Doi: 10.28991/ESJ-2023-07-04-016 Full Text: PD

Gallai-Ramsey and vertex proper connection numbers

Given a complete graph G, we consider two separate scenarios. First, we consider the minimum number N such that every coloring of G using exactly k colors contains either a rainbow triangle or a monochromatic star on t vertices. This number is known for small cases and generalized for larger cases for a fixed k. Second, we introduce the vertex proper connection number of a graph and provide a relationship to the chromatic number of minimally connected subgraphs. Also a notion of total proper connection is introduced and a question is asked about a possible relationship between the total proper connection number and the vertex and edge proper connection numbers