7 research outputs found
Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids
A prismatoid is a polytope with all its vertices contained in two parallel
facets, called its bases. Its width is the number of steps needed to go from
one base to the other in the dual graph. The first author recently showed that
the existence of counter-examples to the Hirsch conjecture is equivalent to
that of -prismatoids of width larger than , and constructed such
prismatoids in dimension five. Here we show that the same is impossible in
dimension four. This is proved by looking at the pair of graph embeddings on a
2-sphere that arise from the normal fans of the two bases.Comment: This paper merges and supersedes the papers arXiv:1101.3050 (of the
last two authors) and arXiv:1102.2645 (of the first author
The width of 5-dimensional prismatoids
Santos' construction of counter-examples to the Hirsch Conjecture (2012) is
based on the existence of prismatoids of dimension d of width greater than d.
Santos, Stephen and Thomas (2012) have shown that this cannot occur in . Motivated by this we here study the width of 5-dimensional prismatoids,
obtaining the following results:
- There are 5-prismatoids of width six with only 25 vertices, versus the 48
vertices in Santos' original construction. This leads to non-Hirsch polytopes
of dimension 20, rather than the original dimension 43.
- There are 5-prismatoids with vertices and width for
arbitrarily large . Hence, the width of 5-prismatoids is unbounded.Comment: 31 pages, 10 figures. Changes from v1: the introduction has been
edited, and a minor correction made in the statement of Proposition 1.
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
Spectral Aspects of Cocliques in Graphs
This thesis considers spectral approaches to finding maximum cocliques in graphs. We focus on the relation between the eigenspaces of a graph and the size and location of its maximum cocliques.
Our main result concerns the computational problem of finding the size of a maximum coclique in a graph. This problem is known to be NP-Hard for general graphs. Recently, Codenotti et al. showed that computing the size of a maximum coclique is still NP-Hard if we restrict to the class of circulant graphs. We take an alternative approach to this result using quotient graphs and coding theory. We apply our method to show that computing the size of a maximum coclique is NP-Hard for the class of Cayley graphs for the groups where is any fixed prime.
Cocliques are closely related to equitable partitions of a graph, and to parallel faces of the eigenpolytopes of a graph. We develop this connection and give a relation between the existence of quadratic polynomials that vanish on the vertices of an eigenpolytope of a graph, and the existence of elements in the null space of the Veronese matrix. This gives a us a tool for finding equitable partitions of a graph, and proving the non-existence of equitable partitions. For distance-regular graphs we exploit the algebraic structure of association schemes to derive an explicit formula for the rank of the Veronese matrix. We apply this machinery to show that there are strongly regular graphs whose -eigenpolytopes are not prismoids.
We also present several partial results on cocliques and graph spectra. We develop a linear programming approach to the problem of finding weightings of the adjacency matrix of a graph that meets the inertia bound with equality, and apply our technique to various families of Cayley graphs. Towards characterizing the maximum cocliques of the folded-cube graphs, we find a class of large facets of the least eigenpolytope of a folded cube, and show how they correspond to the structure of the graph. Finally, we consider equitable partitions with additional structural constraints, namely that both parts are convex subgraphs. We show that Latin square graphs cannot be partitioned into a coclique and a convex subgraph