Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors.

I am interested in $2$-colorable triangulations. Easy examples are boundaries of bipyramids over even polygons such as the simplicial comlex $\{123,134,145,152, 623,634,645,652\}$

Question: Are there efficient criteria to tell if a triangulation is $2$-colorable? How to construct $2$-colorable triangulations? Do they arise in some contexts?

I am also interested in triangulations of other manifolds, so any such references are welcome.