Tudor Zamfirescu

A two-connected planar graph without concurrent longest paths

Abstract An example of a two-connected planar graph without concurrent longest paths is provided.

Extreme points of the distance function on convex surfaces

We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus’ conjecture that “almost all” points admit a single farthest point on the surface.

Nearly all convex bodies are smooth and strictly convex

By an old result of Klee, those convex bodies which are not smooth or not strictly convex form a set of first Baire category. It is proved here that they are “even fewer”: they only form a σ-porous set.

The nearest point mapping is single valued nearly everywhere

0. Introduction. The function p r : l R d ~ 2 ~d called metric projection or nearest point mapping is well-known: Fo r a given closed set K c IR a, p r associates to each x ~ ]R d the set of all points of K closest to x. It is known that PK is single valued almost everywhere and at most points of p d ([2]), i.e. PK is not single valued on a set of measure zero and first Baire category. We shall prove here that PK is single valued nearly everywhere, i.e. PK is not single valued on a a -porous set, which implies both preceding assertions. We also establish that, for most compact sets K, PK is not single valued at densely many points. This will not happen, however, if the boundary of K is smooth enough, as we shall see in the last section.

Hamiltonian properties of grid graphs

This paper presents sufficient conditions for a grid graph to be Hamiltonian. It is proved that all finite grid graphs of positive width have Hamiltonian line graphs.

Hamiltonian properties of Toeplitz graphs

Conditions are given for the existence of hamiltonian paths and cycles in the so-called Toeplitz graphs, i.e. simple graphs with a symmetric Toeplitz adjacency matrix.

Farthest points on convex surfaces

In the very enjoyable book ([3], p. 44) of Craft, Falconer and Guy we read: “We take... a... convex surface C in R3.... Steinhaus... asked... what can be said qualitatively about the set of all “farthest points” from a point x .” Our aim here is to investigate this question. Let S be the space of all closed convex surfaces (i.e. boundaries of open bounded convex sets) in R3 and denote, for any S ∈ S and x ∈ S , by Fx the set of all farthest points from x and by Cx the set of all points joined with x by at least two segments (i.e. shortest paths). We shall see that to any point x on a closed convex surface we may associate in a natural way a point or a Jordan arc Jx ⊃ Fx lying in the cut locus of x . This will provide the topological characterization of Fx . Further we shall prove an easily stated, remarkable geometric property of Fx : any three of its points (if it contains at least three) form an obtuse or right geodesic triangle. Moreover, the preceding result is shown to hold for a large class of geodesic triangles with vertices in the cut locus. Interestingly, the extreme case of a right triangle does not imply the degeneracy of the surface. For the reader, to be familiar with Aleksandrov’s book [1] (see also [2), [4]) would be of considerable help. The usual intrinsic metric of S ∈ S , induced by the Euclidean distance in R3 will be denoted by ρ. Let x ∈ S , and denote by Ex the cut locus of x , i.e. the set of all endpoints different from x of maximal (by inclusion) segments starting at x . Also, let E be the set of all endpoints of S , i.e. points not interior to any segment of S . The set

Acute triangulations of the regular dodecahedral surface

In this paper we consider geodesic triangulations of the surface of the regular dodecahedron. We are especially interested in triangulations with angles not larger than @p/2, with as few triangles as possible. The obvious triangulation obtained by taking the centres of all faces consists of 20 acute triangles. We show that there exists a geodesic triangulation with only 10 non-obtuse triangles, and that this is best possible. We also prove the existence of a geodesic triangulation with 14 acute triangles, and the non-existence of such triangulations with less than 12 triangles.

Intersecting longest paths and longest cycles: A survey

This is a survey of results obtained during the last 45 years regarding the intersection behaviour of all longest paths, or all longest cycles, in connected graphs. Planar graphs and graphs of higher connectivity receive special attention. Graphs embeddable in the cubic lattice of arbitrary dimension, and graphs embeddable in the triangular or hexagonal lattice of the plane are also discussed. Results concerning the case when not all, but just some longest paths or cycles are intersected, for example two or three of them, are also reported.

Multiple farthest points on Alexandrov surfaces

The farthest point mapping on compact surfaces, associating to each point x of the surface the set of absolute maxima of the intrinsic distance from x, is for some surfaces single- valued and a homeomorphism, while for other surfaces it is not single-valued, and not surjective. These two big classes are not very well understood. For instance it is still unknown whether, say in the convex case, the second class is dense. For a C 2 metric on both surfaces and the space of surfaces, the first class has, however, nonempty interior. We describe various properties of the sets of critical points, and of relative and absolute maxima of distance functions, and find several connections between them. We see for example that, on smooth surfaces homeomorphic to S 2 , a point cannot be critical with respect to more than one other point. Sufficient conditions for a surface to belong to the second class will be formulated and a particular Tannery surface belonging to the boundary of both classes will be presented.