List-coloring and sum-list-coloring problems on graphs

Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained. A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs. We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure. Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Supervisory Control Theory for Controlling Swarm Robotics Systems

Swarm robotics systems have the potential to tackle many interesting problems. Their control software is mostly created by ad-hoc development. This makes it hard to deploy swarm robotics systems in real-world scenarios as it is difficult to analyse, maintain, or extend these systems. Formal methods can contribute to overcome these problems. However, they usually do not guarantee that the implementation matches the specification because the system’s control code is typically generated manually. This thesis studies the application of the supervisory control theory (SCT) framework in swarm robotics systems. SCT is widely applied and well established in the man- ufacturing context. It requires the system and the desired behaviours (specifications) to be defined as formal languages. In this thesis, regular languages are used. Regular languages, in the form of deterministic finite state automata, have already been widely applied for controlling swarm robotics systems, enabling a smooth transition from the ad-hoc development currently in practice. This thesis shows that the control code for swarm robotics systems can be automatically generated from formal specifications. Several case studies are presented that serve as guidance for those who want to learn how to specify swarm behaviours using SCT formally. The thesis provides the tools for the implementation of controllers using formal specifications. Controllers are validated on swarms of up to 600 physical robots through a series of systematic experiments. It is also shown that the same controllers can be automatically ported onto different robotics platforms, as long as they offer the required capabilities. The thesis extends and incorporates techniques to the supervisory control theory framework; specifically, the concepts of global events and the use of probabilistic generators. It can be seen as a step towards making formal methods a standard practice in swarm robotics