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Concepts connect the minimum number of passwords needed with the security system are monochromatic connectivity and rainbow connectivity. Rainbow connectivity has been introduced by Chartrand, et al. (2008), and monochromatic connectivity has been introduced by Caro and Yuster (2011). The focus of this paper is G, which is a simple, trivial, undirected, and 4-connected graph. Let s, and n be natural numbers, G is an ordered set of vertices set V(G) and edges set E(G), a 4-connected graph with order n. A coloring function d:E(G)→[1, s] is called rainbow edge s-coloring, if each pair of vertices u and v in V (G) has 4 internally disjoint u-v paths that have different colors or rainbow paths. The minimum number of colors needed so that each pair of vertices u and v in V(G) has 4 rainbow u-v paths called rainbow 4-connection number, rc4(G). Furthermore, the monochromatic u-v path is a u-v path that has the same colors. The monochromatic connection number, cr(G), is the minimum colors needed so that each pair of vertices u and v in V(G) has a monochromatic u-v path. In this paper, we show rc4(G) and cr(G) on some special graphs.