The Crossing Number of Join Product of kth Power of Path Pm with Isolated Vertices and Path Pn

A survey of graphs with known or bounded crossing numbers

“…see Figure 31 in which drawings of G 5 3 ✷S 3 and G 5 4 ✷S 3 are drawn with four crossings, rather than the five suggested by [65]. • In 2016, Hsieh and Lin [74] claimed to have determined the crossing number of the join product of various path powers with discrete graphs and path graphs. All of the claimed results appear to be incorrect, as they all rely on an intermediate result, Lemma 5 of [74], which claims that cr(P m−1 m +D n ) = cr(P m−2 m +D n ) for 3 ≤ m ≤ 6 and n ≥ 1.…”

“…However, this result is incorrect. Note here that in [74], P k m is defined as the k-th power on the path graph with m vertices, rather than m edges. The minimal counterexample can be seen by considering graphs on four vertices.…”