On the CF-Connectedness of the k-Power of Pn

The study of graph connectivity is a central topic in graph theory, with CF-connectedness being a specialized property of interest. A connected graph is CF-connected if, in every optimal drawing, there exists a path between every pair of vertices such that no edges cross. This paper explores the CF-connectedness of the k-power of Pn, denoted Pnk, where Pn is a path on n vertices, Pnk is a graph on the same vertex set as Pn, and an edge {u,v} exists in Pnk if the distance between u and v on Pn is at most k. The paper concludes with a controversial drawing of the graph P105 with only 16 crossings, which refutes the truth of Zheng et al.’s conjecture that the upper bound cr(Pn5)≤4n−23 holds with equality for all n≥8.