Rainbow Connection Number of Shackle Graphs
M. Ali Hasan, Risma Yulina Wulandari, A.N.M. Salman ยท Advances in computer science research ยท 2022
Let ๐บ be a simple, finite and connected graph.For a natural number ๐, we define an edge coloring ๐: ๐ธ(๐บ) โ {1,2, โฆ , ๐} where two adjacent edges can be colored the same.A ๐ข -๐ฃ path (a path connecting two vertices ๐ข and ๐ฃ in ๐(๐บ)) is called a rainbow path if no two edges of path receive the same color.If there exists a ๐ข -๐ฃ rainbow path for any two distinct vertices in ๐(๐บ), then ๐บ is called rainbow connected.In this case, ๐ is called a rainbow ๐ -coloring.The rainbow connection number of G, denoted by ๐๐(๐บ), is the smallest number ๐ such that ๐บ has a rainbow ๐ -coloring.In this paper, we obtain upper and lower bounds of rainbow connection number of shackle graph ๐บ for any graph ๐บ.Furthermore, we show that these bounds are sharp.Then, we get the exact value of rainbow connection number of shackle sun graph, friendship, cycle, complete graph with one edge removed, and fan graph with two certain spokes removed.