An edge-colored graph is rainbow if no two edges of the graph have the same color. An edge-colored graph is proper if every two adjacent edges receive distinct colors. A strongly edge-colored graph is a proper edge-colored graph such that every path of length 3 is rainbow. An edge-colored graph of order n is rainbow k-path-pancyclic if every rainbow k-path is contained in a rainbow l-cycle for each \(\max \{3,k+2\}\le l \le n\) . An edge-colored graph G of order n is rainbow panconnected if for any two vertices a and b of V(G), there exists a rainbow path of length l joining a and b for every integer \(d_G(a, b)\le l \le n-1\) , where \(d_G(a, b)\) denotes the shortest distance between vertices a and b in G. In this paper, we prove that every strongly edge-colored graph with order n and minimum degree \(\delta \ge \frac{2n+k}{3} \) is rainbow k-path pancyclic. We also prove that every strongly edge-colored graph with \(n\ge 10\) vertices and \( \delta \ge \frac{2n+1}{3}\) is rainbow panconnected.
