Graphs In 3-space Research Articles

On coloring box graphs

We consider the chromatic number of a family of graphs we call box graphs, which arise from a box complex in n-space. It is straightforward to show that any box graph in the plane has an admissible coloring with three colors, and that any box graph in n-space has an admissible coloring with n+1 colors. We show that for box graphs in n-space, if the lengths of the boxes in the corresponding box complex take on no more than two values from the set {1,2,3}, then the box graph is 3-colorable, and for some graphs three colors are required. We also show that box graphs in 3-space which do not have cycles of length four (which we call “string complexes”) are 3-colorable.

All toroidal embeddings of polyhedral graphs in 3-space are chiral

We investigate the possibility of forming achiral knottings of polyhedral (3-connected) graphs whose minimal embeddings lie in the genus-one torus. Various analyses to show that all examples are chiral. This result suggests a simple route to forming chiral molecules via templating on a toroidal substrate.

Ravels: knot-free but not free. Novel entanglements of graphs in 3-space

Molecular and extended framework materials, from proteins to catenanes and metal–organic frameworks, can assume knotted configurations in their bonding networks (the chemical graph). Indeed, knot theory and structural chemistry have remained closely allied, due to those connections. Here we introduce a new class of graph entanglement: “ravels”. These ravels—often chiral—tangle a graph without the presence of knots. Just as knots lie within cycles in the graph, ravels lie in the vicinity of a vertex. We introduce various species of ravels, including fragile ravels, composite ravels and shelled ravels. The role of ravels is examined in the context of finite and infinite graphs—analogous to molecular and extended framework nets—related to the diamond net.

The 3-move and knotted 4-valent graphs in 3-space

The 3-move and knotted 4-valent graphs in 3-space

When topology meets chemistry: a topological look at molecular chirality

1. Stereochemical topology 2. Detecting chirality 3. Chiral moebius ladders and related molecular graphs 4. Different types of chirality and achirality 5. Embeddings of complete graphs in 3-space 6. Rigid and non-rigid symmetries of graphs in 3-space 7. Topology of DNA.

Alternating diagrams of 4-regular graphs in 3-space

Embeddings of 4-regular graphs into 3-space are examined by studying graph diagrams, i.e., projections of embedded graphs to an appropriate plane. An invariant of graph diagrams, first introduced by Yokota, is re-formulated and used to show that reduced alternating diagrams have minimal crossing number. The results presented here extend some of the so-called Tait Conjectures.

Topological invariants of graphs in 3-space

Topological invariants of graphs in 3-space

Invariants of theta-curves and other graphs in 3-space

Given a graph in 3-space, in general knotted, can one construct a surface containing the graph in some canonical way so that the embedding of the surface in space, or even the link type of its boundary, is an invariant of the knotted graph? We consider, in particular, surfaces that contain the graph as a spine and that are canonical in the sense of having trivial Seifert linking form. It turns out that θ-curves and K 4-graphs are the unique graphs for which this approach works.

Linkless embeddings of graphs in 3-space

We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of the complement in 3-space of the embedding of every subgraph is free. (ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of K 5 {K_5} or K 3 , 3 {K_{3,3}} . (iii) Any flat embedding of a graph can be transformed to any other flat embedding of the same graph by "3-switches", an analog of 2-switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4-connected graph are either ambient isotopic, or one is ambient isotopic to a mirror image of the other. (iv) A graph has a flat embedding if and only if it has no minor isomorphic to one of seven specified graphs. These are the graphs that can be obtained from K 6 {K_6} by means of Y Δ Y \Delta - and Δ Y \Delta Y -exchanges.

The knot theory of molecules

Macromolecules (such as polymethelyene and DNA) are large and flexible, and can present themselves in 3-space in topologically interesting ways. The branch of topology known asknot theory is the mathematical study of flexible graphs in 3-space. Knot theory can be used to quantify and compare the various configurations of large molecules, and to study the various spatial isomers of molecules which have complicated molecular graphs.

POLYGONAL APPROXIMATION OF FUNCTIONS OF TWO VARIABLES

This paper studies the approximation of functions of two variables by functions whose graphs in 3-space are continuous surfaces which are piecewise flat. By analogy with the case of functions of one variable, we naturally call such approximating functions polygonal. Bibliography: 8 titles.