Professor Holton is currently interested in three areas of research. These are in graph theory (graphs consist of vertices some of which are joined by edges), combinatorics related to computer sciece, and mathematics education.
In graph theory he has been interested in cycles for a long while. The main problems here are to try to understand which graphs have cycles that pass though every vertex and the Travelling Salesperson problem. In the latter problem we are interested in finding the best route for a traveller to visit certain sites and return home. These are very difficult problems but progress can be made in offshoots of them. For instance, we know that in certain kinds of graphs there are cycles that go through a specified number of vertices.
The combinatorial problems that Professor Holton is involved in at the moment are related to sorting machines. Think of objects coming in and being sorted in a certain way. Given arbitrary order of input we would like to know what output orders are possible. A number of promising results have been found with machines such as priority queues (where the lowest numbered object in a machine is exited first) and machines (in series) that interchange consecutive inputs.
Problems in education are more difficult than in mathematics as it is almost impossible to prove anything. He has worked on problem-solving and gifted students with a view to improving our ability to help students understand and enjoy mathematics.