Is it true that nonplanar graphs whose chromatic numbers are less than 5 are almost bipartite?
The reason I raise this question is that in this case, the nonplanarity should come from a minor of $K_{3,3}$ since they cannot have a $K_5$ (am I right?)
So it must be almost bipartite with some additional edges, which means that it should have a big max-cut.
Are there some results concerning this?
I don't know what "almost" bipartite means here, but it seems that you can make examples that are quite far from being bipartite.
For example let you graph has node set $\{1,2,3,4\}\times\{1,2,\ldots n\}$ for a sufficiently large $n$, and edges between any two nodes with different first components. This has chromatic number $4$ and is non-planar (since it has $K_5$ as a minor) for $n\ge 2$