Branko Grünbaum

Acyclic colorings of planar graphs

A coloring of the vertices of a graph byk colors is called acyclic provided that no circuit is bichromatic. We prove that every planar graph has an acyclic coloring with nine colors, and conjecture that five colors are sufficient. Other results on related types of colorings are also obtained; some of them generalize known facts about “point-arboricity”.

An enumeration of simplicial 4-polytopes with 8 vertices

Abstract An enumeration of all the different combinatorial types of 4-dimensional simplicial convex polytopes with 8 vertices is given. It corrects an earlier enumeration attempt by M. Bruckner, and leads to a simple example of a diagram which is not a Schlegel diagram.

Shortness exponents of families of graphs

Abstract Known estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions.

Regular polyhedra—old and new

Although it is customary to define polygons as certain families of edges, when considering polyhedra it is usual to view polygons as 2-dimensional pieces of the plane. If this rather illogical point of view is replaced by consistently understanding polygons as 1-dimensional complexes, the theory of polyhedra becomes richer and more satisfactory. Even with the strictest definition of regularity this approach leads to 17 individual regular polyhedra in the Euclidean 3-space and 12 infinite families of such polyhedra, besides the traditional ones (which consist of 5 Platonic polyhedra, 4 Kepler—Poinsot polyhedra, 3 planar tessellations and 3 Petrie—Coxeter polyhedra). Among the many still open problems that naturally arise from the new point of view, the most obvious one is the question whether the regular polyhedra found in the paper are the only ones possible in the Euclidean 3-space.

Venn Diagrams and Independent Families of Sets.

Abstract : Motivated by the well known notions from probability and logic, the author says that a family of n simple closed curves (A sub 1),...,(A sub n) in the Euclidean plane is independent provided the intersection (*) (X sub 1)(X sub 2)...(X sub n) is non-empty whenever each set (X sub j) is either the interior or else the exterior of (A sub j). An independent family is a Venn diagram if each intersection (*) is connected. These notions are examined from the point of view of combinatorial geometry and several results are obtained; some of them correct erroneous assertion found in the literature. (Modified author abstract)

Pick's theorem

Some years ago, the Northwest Mathematics Conference was held in Eugene, Oregon. To add a bit of local flavor, a forester was included on the program, and those who attended his session were introduced to a variety of nice examples which illustrated the important role that mathematics plays in the forest industry. One of his problems was concerned with the calculation of the area inside a polygonal region drawn to scale from field data obtained for a stand of timber by a timber cruiser. The standard method is to overlay a scale drawing with a transparency on which a square dot pattern is printed. Except for a factor dependent on the relative sizes of the drawing and the square grid, the area inside the polygon is computed by counting all of the dots fully inside the polygon, and then adding half of the number of dots which fall on the bounding edges of the polygon. Although the speaker was not aware that he was essentially using Picks formula, I was delighted to see that one of my favorite mathematical results was not only beautiful, but even useful. (From DeTemple [1989].)

The orchard problem

Abstract : The geometric version of the problem of Kirkman-Steiner triples may be formulated as follows: What is the maximal possible number t(p) of lines each of which contains precisely three points of a suitable set of p points in the Euclidean plane. The first general results were announced by J. J. Sylvester in 1867 and 1868, but up to now no proof of his best claims was published. The authors present a proof of a theorem improving those given by Sylvester, together with several related results. The general estimate they obtain may be put in the form (p(p-3)/6) + 1 < or = t(p) < or = (p(p-3)/6 + 4p/21). (Author)

SYMMETRY IN MOORISH AND OTHER ORNAMENTS

Abstract An investigation of the Moorish ornaments from the Alhambra (in Granada, Spain) shows that their symmetry groups belong to 13 different crystallographic (wallpaper) classes; this corrects several earlier enumerations and claims. The four classes of wallpaper groups missing in Alhambra (pg, p2, pgg, p3ml) have not been found in other Moorish ornaments, either. But the classification of repeating patterns by their symmetry groups is in many cases not really appropriate—account should be taken of the coloring of the patterns, of their interlace characteristics, etc. This leads to a variety of “symmetry groups”, not all of which have been fully investigated. Moreovr, the “global” approach to repeating ornaments is only of limited applicability, since it does not correspond to the way of thinking of the artisans involved, and does not cover all the possibilities of “local” order. The proper mathematical tools for the study of such structures which are only “locally orderly” remain to be developed.

Vertices missed by longest paths or circuits

Abstract : Let G denote a graph with v(G) vertices, and let c(G) and p(G) denote the maximal numbers of vertices contained in any simple circuit or path in G. The author denotes by c(j,k) (p(j,k)) the greatest lower bounds of v(G) where G is a k-connected graph such that for every choice of j vertices of G there exists in G a simple circuit (path) with c(G) (p(G)) vertices that misses the j chosen vertices. Analogously defined numbers with G restricted to planar graphs are denoted by (c sub 0)(j,k) and (p sub 0)(j,k). Extending various known results it is established that (c sub 0)(1,3) < or = 124, (p sub 0)(1,3) < or = 484, c(2.3) < or = 90 and p(2.3) < or = 324. (Modified author abstract)

Some semicontinuity theorems for convex polytopes and cell-complexes

In a finite-dimensional EUCLIDean space E , a convex polytope P is a set which is the convex hull of a finite set; here such a set P will simply be called a polytope, or, if it is d-dimensional, a d-polytope. 2) An s-/ace of P is an s-dimensional set (necessarily an s-polytope) which is either P itself or is the intersection of P with a supporting hyperplane. The number of s-faces of P will be denoted by [~(P) and the s-measure of their union by ~ ( P ) . Our section headings are as follows : 1. The functions /~ for cell-complexes 2. A pulling process 3. The functions