Groups & Graphs 3.5 for Macintosh

Last update=14 Jul, 2009


G&G 3.6 for OS X is now available. Go to the download or news page for further info.

New in Version 3.5 (Jan, 2009)

Projective Maps

G&G can now test graphs for projective planarity, and embed them in the projective plane.

Hexagon Maps

G&G can now represent the torus as either a rectangle (the usual model) or as a hexagon. The hexagon model shows different symmetries of the torus. Some graphs look quite different in the two models of the torus.

4D Polyhedra

Some predefined 4D Polyhedra can be constructed (5-cell, 8-cell, 16-cell). 4D Polyhedra can also be constructed from 3D Polyhedra. A 4D Polyhedron is projected into 3D in order to display it. Some 4D rotations can be displayed.

Generation of Trees

G&G can generate several kinds of trees: rooted trees, unrooted trees, central trees, and bicentral trees, on n vertices.

n-Body Simulations

Given n >= 2 "planets" interacting via a Newtonian gravitational force, the orbits of the planets can be displayed. Nearly all n-body systems exhibit chaotic behaviour when n >= 3.


New in Version 3.2 (May, 2006)

Projective Maps

There is now support for Projective Maps, ie, graphs drawn on the projective plane. A number of operations are available for projective maps: find dual graph, medial digraph, double cover, construct the automorphism group of the embedding, truncate a graph. An algorithm to determine whether an arbitrary graph is projective has not yet been implemented. This will be available in a future release.

Polyhedra and Convex Hulls

There is now support for 3D Polyhedra. A number of predefined polyhedra are available. Polyhedra can be rotated, truncated, and their duals can be constructed. Convex Hulls in 3D can be computed. The result is displayed as a polyhedron.


There is now support for a variety of Fractals. The Mandelbrot set and a number of Julia sets in the complex plane can be computed.

Text I/O

There is increased support for exporting and importing graphs and other combinatorial objects via ascii text files, or the clipboard.


New in Version 3.0

Torus Maps

There is now support for Torus Maps, ie, graphs drawn on the torus. A number of operations are available for torus maps: find dual graph, medial digraph, construct the automorphism group of the embedding, truncate a graph, or apply torus homeomorphisms --> rotate or twist torus. There is currently no algorithm implemented to determine whether an arbitrary graph is toroidal.

Graph Layout

The graph layout algorithm for planar graphs and torus maps has been improved.

Hamilton Cycles

It is now possible to find all hamilton cycles in a graph, and write them to a file. Each ham cycle can be encoded as a graph, so that ham cycles equivalent under an automorphism of the graph can be detected.

Graph Matrices

The Adjacency matrix and Kirchhoff matrix of a graph can now be constructed and placed on the clipboard for export to other programs.

Random Graphs

Random graphs can now be required to be planar triangulations. Random subgraphs of a graph can also be constructed.

Incidence Graphs

Given an arbitrary connected bipartite graph with girth at least 6, G&G will treat it as the incidence graph of a projective configuration, and attempt to draw it in the projective plane. This can always be done for graphs with maximum degree 3.

File Structure

A new file structure for saved graphs, digraphs, and groups has been implemented. The Mac version and Windows version now can read and write the same files.


Normalizers, Centralizers, and the Center of a group can now be calculated.

Group Products

Wreath products, direct products and subdirect products of groups can now be constructed.

Block Systems

When looking for a block system in a group, G&G will now ask for a prime required to be a divisor of the block size.

Batch Isomorphism

A memory leak in Batch Aut(X) has been corrected. It is now possible to use Batch Aut(X) on files containing hundreds of thousands of graphs. G&G will construct a file of certificates for the graphs, sort it, and select a representative of each isomorphism type.

Bug Fixes

A number of minor bugs have been corrected.


Note: The Blue&White G3's and the older G4's contain a bug in the Quickdraw line drawing algorithms. Oblique lines of thickness 2 or more pixels drawn in XOR mode do not draw properly when both ends are in the window's clipping region. G&G uses this method to draw subgraphs. Consequently bold subgraphs will not display properly on these Macs. This bug does not affect the older beige G3's or earlier Macs.


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