Group Sum Chromatic Number of Graphs

We investigate the \textit{group sum chromatic number} (\gchi(G)) of graphs, i.e. the smallest value $s$ such that taking any Abelian group \gr of order $s$, there exists a function f:E(G)\rightarrow \gr such that the sums of edge labels properly colour the vertices. It is known that \gchi(G)\in\{\chi(G),\chi(G)+1\} for any graph $G$ with no component of order less than $3$ and we characterize the graphs for which \gchi(G)=\chi(G).Comment: Accepted for publication in European Journal of Combinatorics, Elsevie

Group Irregularity Strength of Connected Graphs

We investigate the group irregularity strength ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group \gr of order $s$, there exists a function f:E(G)\rightarrow \gr such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph $G$ of order at least 3, $s_g(G)=n$ if $n\neq 4k+2$ and $s_g(G)\leq n+1$ otherwise, except the case of some infinite family of stars

Research Problems from the BCC21

AbstractA collection of open problems, mostly presented at the problem session of the 21st British Combinatorial Conference

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

This Week's Finds in Mathematical Physics (1-50)

These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).Comment: 242 page

Continuous automata: bridging the gap between discrete and continuous time system models

The principled use of models in design and maintenance of a system is fundamental to the engineering methodology. As the complexity and sophistication of systems increase so do the demands on the system models required to design them. In particular the design of agent systems situated in the real world, such as robots, will require design models capable of expressing discrete and continuous changes of system parameters. Such systems are referred to as mode-switching or hybrid systems.This thesis investigates ways in which time is represented in automata system models with discretely and continuously changing parameters. Existing automaton approaches to hybrid modelling rely on describing continuous change at a sequence of points in time. In such approaches the time that elapses between each point is chosen non- deterministically in order to ensure that the model does not over-step a discrete change. In contrast, the new approach this thesis proposes describes continuous change by a continuum of points which can naturally and deterministically capture such change. As well as defining the semantics of individual models the nature of the temporal representation is particularly important in defining the composition of modular com­ponents. This new approach leads to a clear compositional semantics based on the synchronization of input and output values.The main contribution of this work is the derivation of a limiting process which provides a theoretical foundation for this new approach. It not only provides a link between dis­crete and continuous time representations, but also provides a basis for deciding which continuous time representations are theoretically sound. The resulting formalism, the Continuous I/O machine, is demonstrated to be comparable to Hybrid Automata in expressibility, but its representation of time gives it a much stronger compositional semantics based on the discrete synchronous machines from which it is derived.TThe conclusion of this work is that it is possible to define an automaton model that describes a continuum of events and that this can be effectively used to model complete mode-switching physical systems in a modular fashion