Nicolas Van Cleemput

Generation of various classes of trivalent graphs

It turns out that there exist numerous useful classes of cubic graphs. Some are needed in connection with maps, hypermaps, configurations, polytopes, or covering graphs. In this paper, we briefly explore these connections and give motivation why some classes of cubic graphs should be generated. Then we describe the algorithms we used to generate these classes. The results are presented in various tables.

10-Gabriel graphs are Hamiltonian

We prove that the 10-Gabriel graph of any set of points is Hamiltonian.We discuss a possible way to further improve upon the previous result.We show that there exist sets of points whose 1-Gabriel graph is not Hamiltonian. Given a set S of points in the plane, the k-Gabriel graph of S is the geometric graph with vertex set S, where p i , p j ? S are connected by an edge if and only if the closed disk having segment p i p j ? as diameter contains at most k points of S ? { p i , p j } . We consider the following question: What is the minimum value of k such that the k-Gabriel graph of every point set S contains a Hamiltonian cycle? For this value, we give an upper bound of 10 and a lower bound of 2. The best previously known values were 15 and 1, respectively.

On the Strongest Form of a Theorem of Whitney for Hamiltonian Cycles in Plane Triangulations

In this article, we investigate hamiltonian cycles in plane triangulations. The aim of the article is to find the strongest possible form of Whitneys theorem about hamiltonian triangulations in terms of the decomposition tree defined by separating triangles. We will decide on the existence of nonhamiltonian triangulations with given decomposition trees for all trees except trees with exactly one vertex with degree k{4,5} and all other degrees at most 3. For these cases, we show that it is sufficient to decide on the existence of nonhamiltonian triangulations with decomposition tree K-1,K- 4 or K-1,K- 5. We also give computational results on the size of a possible minimal nonhamiltonian triangulation with these decomposition trees.

On the number of hamiltonian cycles in triangulations with few separating triangles

In this article, we investigate the number of hamiltonian cycles in triangulations. We improve a lower bound of |V|/log2|V| for the number of hamiltonian cycles in triangulations without separating triangles (4-connected triangulations) by Hakimi, Schmeichel, and Thomassen to a linear lower bound and show that a linear lower bound even holds in the case of triangulations with one separating triangle. We confirm their conjecture about the number of hamiltonian cycles in triangulations without separating triangles for up to 25 vertices and give computational results and constructions for triangulations with a small number of hamiltonian cycles and 1–5 separating triangles.

Construction of planar 4-connected triangulations

In this article we describe a recursive structure for the class of 4-connected triangulations or – equivalently – cyclically 4-connected plane cubic graphs.

Hamiltonian properties of polyhedra with few 3-cuts : a survey

Abstract We give an overview of the most important techniques and results concerning the hamiltonian properties of planar 3-connected graphs with few 3-vertex-cuts. In this context, we also discuss planar triangulations and their decomposition trees. We observe an astonishing similarity between the hamiltonian behavior of planar triangulations and planar 3-connected graphs. In addition to surveying, (i) we give a unified approach to constructing non-traceable, non-hamiltonian, and non-hamiltonian-connected triangulations, and show that planar 3-connected graphs (ii) with at most one 3-vertex-cut are hamiltonian-connected, and (iii) with at most two 3-vertex-cuts are 1-hamiltonian, filling two gaps in the literature. Finally, we discuss open problems and conjectures.

Automated Conjecturing I: Fajtlowicz's Dalmatian Heuristic Revisited (Extended Abstract)

This condensed summary highlights the results of a 2016 AIJ paper reporting on a successful generalpurpose conjecturing program.

Sizes of pentagonal clusters in fullerenes

Stability and chemistry, both exohedral and endohedral, of fullerenes are critically dependent on the distribution of their obligatory 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes and that the pentagons in these fullerenes can be at an arbitrarily large distance from each other. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1. In this paper we study the combinations of cluster sizes that can occur in fullerenes and whether the clusters can be at an arbitrarily large distance from each other. For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene, and state whether the different clusters can be at an arbitrarily large distance from each other. We will prove that all partitions with largest cluster of size 5 or less can occur in an infinite number of fullerenes with the clusters at an arbitrarily large distance of each other, that 9 partitions occur in only a finite number of fullerene isomers and that 15 partitions do not occur at all in fullerenes.

Spherical tilings by congruent quadrangles: Forbidden cases and substructures

In this article we show the non-existence of a class of spherical tilings by congruent quadrangles. We also prove several forbidden substructures for spherical tilings by congruent quadrangles. These are results that will help to complete of the classification of spherical tilings by congruent quadrangles.

Alternating plane graphs

A plane graph is called alternating if all adjacent vertices have different degrees, and all neighboring faces as well. Alternating plane graphs were introduced in 2008. This paper presents the previous research on alternating plane graphs. There are two smallest alternating plane graphs, having 17 vertices and 17 faces each. There is no alternating plane graph with 18 vertices, but alternating plane graphs exist for all cardinalities from 19 on. From a small set of initial building blocks, alternating plane graphs can be constructed for all large cardinalities. Many of the small alternating plane graphs have been found with extensive computer help. Theoretical results on alternating plane graphs are included where all degrees have to be from the set {3,4,5}. In addition, several classes of “weak alternating plane graphs” (with vertices of degree 2) are presented.