On Rainbow k-Connection Number of Special Graphs and It's Sharp Lower Bound

International Conference on Mathematics: Education, Theory and Application

数学;教育

Agustin, Ika Hesti^1,2 ; Dafik^1,3 ; Gembong, A.W.^1,2 ; Alfarisi, Ridho^1,4

CGANT, University of Jember, Indonesia^1

Mathematics Depart., University of Jember, Indonesia^2

Mathematics Edu. Depart., University of Jember, Indonesia^3

Department of Mathematics, Sepuluh Nopember Institute of Technology, Surabaya, Indonesia^4

关键词: Connected graph;    Connection number;    Edge colored graphs;    Graph G;    Lower bounds;    Special Graphs;    Undirected graph;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/855/1/012003/pdf
DOI  :  10.1088/1742-6596/855/1/012003

学科分类：发展心理学和教育心理学

来源: IOP

【 摘 要 】

Let G = (V, E) be a simple, nontrivial, finite, connected and undirected graph. Let c be a coloring c : E(G) → {1, 2, , s}, s ∈ N. A path of edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph G is said to be a rainbow connected graph if there exists a rainbow u - v path for every two vertices u and v of G. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of k colors required to edge color the graph such that the graph is rainbow connected. Furthermore, for an l-connected graph G and an integer k with 1 ≤ k ≤ l, the rainbow k-connection number rck(G) of G is defined to be the minimum number of colors required to color the edges of G such that every two distinct vertices of G are connected by at least k internally disjoint rainbow paths. In this paper, we determine the exact values of rainbow connection number of some special graphs and obtain a sharp lower bound.