Mathematics – Combinatorics
Scientific paper
2010-06-30
Mathematics
Combinatorics
11 pages, 1 figure; minor revisions; final version to appear in Beitr. Alg. Geom
Scientific paper
10.1007/s13366-011-0064-4
We study the \emph{picture space} $X^d(G)$ of all embeddings of a finite graph $G$ as point-and-line arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural decomposition into smooth quasiprojective subvarieties called \emph{cellules}, indexed by partitions of the vertex set of $G$, and the irreducible components of $X^d(G)$ correspond to cellules that are maximal with respect to a partial order on partitions that is in general weaker than refinement. We study both general properties of this partial order and its characterization for specific graphs. Our results include complete combinatorial descriptions of the irreducible components of the picture spaces of complete graphs and complete multipartite graphs, for any ambient dimension $d$. In addition, we give two graph-theoretic formulas for the minimum ambient dimension in which the directions of edges in an embedding of $G$ are mutually constrained.
Enkosky Thomas
Martin Jeremy L.
No associations
LandOfFree
Graph Varieties in High Dimension does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Graph Varieties in High Dimension, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Graph Varieties in High Dimension will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-154023