Joseph Zaks

Geometric permutations for convex sets

Abstract A geometric permutation is the pair of permutations formed by a common transversal for a finite family of disjoint convex sets in the plane. It is shown that if A is a family of n convex and pairwise disjoint segments in the plane then the number of geometric permutations formed by all common transversals of A cannot be more than n .

Extending Kotzig’s theorem

AbstractThe weight of a graphG is the minimum sum of the two degrees of the end points of edges ofG. Kotzig proved that every graph triangulating the sphere has weight at most 13, and Grünbaum and Shephard proved that every graph triangulating the torus has weight at most 15. We extend these results for graphs, multigraphs and pseudographs “triangulating” the sphere withg handlesSg,g≧1, showing that the corresponding weights are at most about

Non-Hamiltonian non-Grinbergian graphs

\sqrt {48g} ,8g + 7

Pairs of Hamiltonian circuits in 5-connected planar graphs

and 24g−9, respectively; if a (multi, pseudo) graph triangulatesSg and it is big enough, then its weight is at most 15.

Convexification of polygons by flips and by flipturns

Abstract Consider a rectangular art gallery divide into n rectangular rooms, such that any two rooms sharing a wall in common have a door connecting them. How many guards need to be stationed in the gallery so as to protect all of the rooms in our gallery? Note that of a guard is stationed at a door, he will be able to guard two rooms. Our main aim in this paper is to show that ⌈ n /2⌉ guards are always sufficient to protect all rooms in a rectangular art gallery. Extensions of our result are obtained for non-rectangular galleries and for 3-dimensional art galleries.

On illuminating line segments in the plane

Abstract Settling a question of Tutte and a similar question of Grunbaum and Zaks, we present a 3-valent 3-connected planar graph that has only pentagons and octagons, has 92 (200, respectively) vertices and its longest circuit (path, respectively) contains at most 90 (198, respectively) vertices; moreover, it is shown that the family of all 3-valent 3-connected planar graphs, having n -gons only if n ≡ 2 (mod3) (or n ≡ 1 (mod3)), has a shortness exponent which is less than one.

Nearly-neighborly families of Tetrahedra and the decomposition of some multigraphs

Given a setA inR2 and a collectionS of plane sets, we say that a lineL separatesA fromS ifA is contained in one of the closed half-planes defined byL, while every set inS is contained in the complementary closed half-plane.We prove that, for any collectionF ofn disjoint disks inR2, there is a lineL that separates a disk inF from a subcollection ofF with at least ⌌(n−7)/4⌍ disks. We produce configurationsHn andGn, withn and 2n disks, respectively, such that no pair of disks inHn can be simultaneously separated from any set with more than one disk ofHn, and no disk inGn can be separated from any subset ofGn with more thann disks.We also present a setJm with 3m line segments inR2, such that no segment inJm can be separated from a subset ofJm with more thanm+1 elements. This disproves a conjecture by N. Alonet al. Finally we show that ifF is a set ofn disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a lineL that separates one of the segments from at least ⌌n/3⌍+1 elements ofF.

Arbitrarily large neighborly families of symmetric convex polytopes

Abstract Settling a problem raised by B. Grunbaum, J. Malkevitch, and the author, we present 5-valent 5-connected planar graphs that admit no pairs of edgedisjoint Hamiltonian circuits; our smallest example has 176 vertices. This is used to construct an infinite family of 5-valent 5-connected planar graphs, in which every member has the property that any pair of Hamiltonian circuits in it share at least about 1 168 of their edges. We construct 4- and 5-valent, 3-connected non-Hamiltonian planar graphs.