# Thickness of non-complete bipartite graph [closed]

Is there any analytical result or algorithm on computing the thickness of a non-complete bipartite graph?

I really tried to find anything but didn't succeed.

This problem is NP-hard for general graphs. Also, if you have an arbitrary graph $$G$$, if you subdivide each edge, you get a bipartite graph $$H$$ with the same thickness.
Proof. Any decomposition of $$G$$ into planar subgraphs corresponds to a decomposition of $$H$$ in which both halves of every subdivided edge are in the same planar subgraph. Call such a decomposition of $$H$$ "canonical". Every canonical decomposition of $$H$$ corresponds to a decomposition of $$G$$. Meanwhile, if you have a non-canonical decomposition of $$H$$, you can take any subdivided edge whose halves are in different planar subgraph, and move one half to join the other: this does not destroy planarity. We get a canonical decomposition with the same number of parts. So the only decompositions worth considering are canonical, and since those correspond to decompositions of $$G$$, the thickness of $$H$$ is the same as the the thickness of $$G$$.