A Note on Nonplanar Sequences of Iterated Jump Graphs

For a graph \(G\), the jump graph \(J(G)\) is that graph whose vertices are the edges of \(G\) and where two vertices of \(J(G)\) are adjacent if the corresponding edges are not adjacent. For \(k \geq 2\), the \(k\)th iterated jump graph \(J^k(G)\) is defined as \(J(J^{k-1}(G))\), where \(J^1(G) = J(G)\). An infinite sequence \(\{G_i\}\) of graphs is planar if every graph \(G_i\) is planar; while the sequence \(\{G_i\}\) is nonplanar otherwise. All connected graphs \(G\) for which \(\{J^k(G)\}\) is planar have been determined. In this paper, we investigate those connected graphs \(G\) for which \(\{J^k(G)\}\) is nonplanar. It is shown that if \(\{J^k(G)\}\) is a nonplanar sequence, then \(J^k(G)\) is nonplanar for all \(k \geq 4\). Furthermore, there are only six connected graphs \(G\) for which \(\{J^k(G)\}\) is nonplanar and \(J^3(G)\) is planar.