17,559 research outputs found
Local Properties via Color Energy Graphs and Forbidden Configurations
The local properties problem of Erdős and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants, by extending the color energy technique---a variant of the additive energy technique from additive combinatorics (color energy was originally introduced by the last two authors [C. Pohoata and A. Sheffer, Combinatorica, 39 (2019), pp. 705-714]). We generalize the concept of color energy to higher color energies and combine these with bounds on the extremal numbers of even cycles. Let f(n,k,l) denote the minimum number of colors required to color the edges of K_n such that every k vertices span at least l colors. It can be easily shown that f(n,k,(k/2)–[k/2]+2 = Θ(n²). Erdős and Gyárfás asked what happens when l = (k/2)-[k/2]+1, one away from the easy case, and derived the bound f(n,k,l) = Ω(n^(4/3)). Our technique significantly improves this to f(n,k,(k/2))–[k/2]+1) = Ω(n^(2-8/k))
Excluding subdivisions of bounded degree graphs
Let be a fixed graph. What can be said about graphs that have no
subgraph isomorphic to a subdivision of ? Grohe and Marx proved that such
graphs satisfy a certain structure theorem that is not satisfied by graphs
that contain a subdivision of a (larger) graph . Dvo\v{r}\'ak found a
clever strengthening---his structure is not satisfied by graphs that contain a
subdivision of a graph , where has "similar embedding properties" as
. Building upon Dvo\v{r}\'ak's theorem, we prove that said graphs
satisfy a similar structure theorem. Our structure is not satisfied by graphs
that contain a subdivision of a graph that has similar embedding
properties as and has the same maximum degree as . This will be
important in a forthcoming application to well-quasi-ordering
The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices
In this paper we report on the full classification of Dirichlet-Voronoi
polyhedra and Delaunay subdivisions of five-dimensional translational lattices.
We obtain a complete list of affine types (L-types) of Delaunay
subdivisions and it turns out that they are all combinatorially inequivalent,
giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra.
Using a refinement of corresponding secondary cones, we obtain
contraction types. We report on details of our computer assisted enumeration,
which we verified by three independent implementations and a topological mass
formula check.Comment: 16 page
Intersection Graphs of L-Shapes and Segments in the Plane
An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: ⌊,⌈,⌋ and ⌉. A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an ⌊, an ⌊ or ⌈, a k-bend path, or a segment, then this graph is called an ⌊-graph, ⌊,⌈-graph, B k -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365–372, 1992], stating that every ⌊,⌈-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are ⌊-graphs, or B k -VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are ⌊-graphs. Furthermore we show that all complements of planar graphs are B 19-VPG-graphs and all complements of full subdivisions are B 2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once
- …