Proof of Lemma 6.3 in ``The crossing number of $K_{4,n}$ on the torus and the Klein bottle"

TL;DR

This paper identifies an error in a previously published lemma regarding the crossing number of complete bipartite graphs on certain surfaces, highlighting the need for correction in that area of graph theory.

Contribution

The authors demonstrate that Lemma 6.3 in the referenced work is incorrect, challenging existing assumptions about crossing numbers on the torus and Klein bottle.

Findings

Lemma 6.3 is invalid as stated

Implications for crossing number calculations

Need for revised proofs or corrections

Abstract

We found that Lemma 6.3 in the paper ``The crossing number of $K_{4,n}$ on the torus and the Klein bottle" is wrong.