On some points-and-lines problems and configurations

We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.Comment: 14 pages, numerous figures of point-and-line configurations; to appear in the Bezdek-50 special issue of Periodica Mathematica Hungaric

Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, $q$, the number of lights on each robot, $k$, and the number of consecutive lights the camera can see, $\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that it is equivalent to finding the maximum number of disjoint $k$-cycles in the de Bruijn graph $\text{dB}(q,\ell)$. We provide several existence results that give the maximum number of cycles in $\text{dB}(q,\ell)$ in various cases. For example, we give an optimal solution when $k=q^{\ell-1}$. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if $\text{dB}(q,\ell)$ can be partitioned into $k$-cycles, then $\text{dB}(q,\ell)$ can be partitioned into $tk$-cycles for any divisor $t$ of $k$. The methods used are based on finite field algebra and the combinatorics of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied Mathematic

Convex Configurations on Nana-kin-san Puzzle

We investigate a silhouette puzzle that is recently developed based on the golden ratio. Traditional silhouette puzzles are based on a simple tile. For example, the tangram is based on isosceles right triangles; that is, each of seven pieces is formed by gluing some identical isosceles right triangles. Using the property, we can analyze it by hand, that is, without computer. On the other hand, if each piece has no special property, it is quite hard even using computer since we have to handle real numbers without numerical errors during computation. The new silhouette puzzle is between them; each of seven pieces is not based on integer length and right angles, but based on golden ratio, which admits us to represent these seven pieces in some nontrivial way. Based on the property, we develop an algorithm to handle the puzzle, and our algorithm succeeded to enumerate all convex shapes that can be made by the puzzle pieces. It is known that the tangram and another classic silhouette puzzle known as Sei-shonagon chie no ita can form 13 and 16 convex shapes, respectively. The new puzzle, Nana-kin-san puzzle, admits to form 62 different convex shapes

Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

The general position number and the iteration time in the P3 convexity

In this paper, we investigate two graph convexity parameters: the iteration time and the general position number. Harary and Nieminem introduced in 1981 the iteration time in the geodesic convexity, but its computational complexity was still open. Manuel and Klav\v{z}ar introduced in 2018 the general position number of the geodesic convexity and proved that it is NP-hard to compute. In this paper, we extend these parameters to the P3 convexity and prove that it is NP-hard to compute them. With this, we also prove that the iteration number is NP-hard on the geodesic convexity even in graphs with diameter two. These results are the last three missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and P3 convexities

Abstractions and Analyses of Grid Games

In this paper, we define various combinatorial games derived from the NQueens Puzzle and scrutinize them, particularly the Knights Game, using combinatorial game theory and graph theory. The major result of the paper is an original method for determining who wins the Knights Game merely from the board\u27s dimensions. We also inspect the Knights Game\u27s structural similarities to the Knight\u27s Tour and the Bishops Game, and provide some historical background and real-world applications of the material

Survey of Graph Embeddings into Compact Surfaces

A prominent question of topological graph theory is what type of surface can a nonplanar graph be embedded into? This thesis has two main goals. First to provide a necessary background in topology and graph theory to understand the development of an embedding algorithm. The main purpose is developing and proving a direct constructive embedding algorithm that takes as input the graph with a particular order of edges about each vertex. The embedding algorithm will not only determine which compact surface the graph can be embedded into, but also determines the particular embedding of the graph on the surface. The embedding algorithm is then used to investigate surfaces into which trees and a class of the complete bipartite graphs can be embedded. Further, the embedding algorithm is used to investigate non-surface separating graph embeddings

On the Vertex Position Number of Graphs

In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set} if for any $y \in S$ the shortest $x,y$-paths in $G$ contain no point of $S\setminus \{ y\}$. We investigate the largest and smallest orders of maximum $x$-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.Comment: A new author added. A result on Kneser graphs has been inserted and the bound for vp^- for triangle-free graphs correcte