To make the question (hopefully) clearer, I reformulated it:
Some triangulation $T$ of a smooth manifold $M$ is a piecewise linear manifold, because smooth manifolds are topological manifolds. Such a triangulation is homeomorphic to the smooth manifold, with some homeomorphism $h: T\rightarrow M$. A PL manifold is defined by the property, that the link of any simplex is a PL sphere or in other words the link is homeomorphic to a sphere. If one takes the 1-skeleton of $T$, one gets a graph $G$. I am curious about the equivalent properties of such a full simplicial complex $T$ and its 1-skeleton $G$:
- Are $G$ and $T$ basically equivalent? In general $T$ should be reconstructible from $G$ (since the simplices form cliques).
- If so, what properties of $G$ are equivalent to the fact, that the link of any simplex in $T$ is homeomorphic to a sphere?
- Aren't PL manifolds with that property automatically closed? As far as I understand, the link of a vertex on the boundary of a triangulation cannot be a PL sphere.
- If so, triangulations of smooth manifolds with a boundary would be excluded. If one wants to include triangulations of such manifolds, what would be the defining property of the PL manifold?
These questions raised in the context of coloring the vertexes of some $T$, to get independend sets of them. If the 1-skeleton is equivalent to some certain type of graph, it may be possible to use optimal algorithms from graph theory.