Action graphs and covering graph techniques

This Project will develop and employ a toolbox of powerful new methods in discrete mathematics in connection with group theory, discrete geometry, algebraic topology, which will be used for attacking several outstanding open problems. In particular a theory of action graphs and further advance of the covering graph techniques will be developed. Generalized action graphs, as we define them, generalize a number of important mathematical concepts such as Cayley color graphs of groups, monodromy groups of maps and hyper-maps on surfaces, permutation-involution description of oriented maps on surfaces, abstract polytopes and maniplexes. For us, an action graph is a finite set of flags, endowed with a collection of permutations and another collection of involutions. By using regular covering projections we obtain the so-called symmetry type graphs that capture the essence of symmetries of the original action graphs. This, combined with our recent theory of representation of graphs, will be a powerful tool for studying geometric and topological realizability questions of certain combinatorial objects. Also, we started a theory of arc types for vertex-transitive graphs, which we intend to apply towards the solution of the poly-circulant conjecture. The Project will be conducted within the following work packages (WP). (WP1) Hadwiger-Nelson problem, alias chromatic number of the Euclidean plane, and unit distance graphs. (WP2) Polycirculant conjecture. This long standing conjecture states that every vertex-transitive graph admits a semi-regular automorphism. (WP3) Development of covering graph techniques and generalized action graphs with applications. Adam-like conjectures and the CI-groups, bi-circulants and their generalizations.

The Project aims to obtain new original scientific results with publications in SCI journals and with presentations at international scientific conferences. The relevance and impact of the expected results of the research will be reflected through the high level of relevance and influence. For instance, we expect to use quotient graphs and voltage graphs as very condensed representation of a large quantities of certain graphs families stored in specialised databases. Preliminary computations show that over 98% storage can be saved. The obtained results will definitely prove beneficial in achieving a better understanding of subtle connections between group theory, discrete geometry, algebraic topology and certain algorithmic aspects related to these connections. This will have a multiplier effect for other interested researchers in the field of discrete mathematics and theoretical computer science. A systematic approach to the promotion will contribute significantly to the usefulness of the research project results, by increasing the added value, adding social touch, and will lead to a continuation of activities in this area even after the end of the project.

The Project aims to obtain new original scientific results with publications in SCI journals and with presentations at international scientific conferences. The relevance and impact of the expected results of the research will be reflected through the high level of relevance and influence. For instance, we expect to use quotient graphs and voltage graphs as very condensed representation of a large quantities of certain graphs families stored in specialised databases. Preliminary computations show that over 98% storage can be saved. The obtained results will definitely prove beneficial in achieving a better understanding of subtle connections between group theory, discrete geometry, algebraic topology and certain algorithmic aspects related to these connections. This will have a multiplier effect for other interested researchers in the field of discrete mathematics and theoretical computer science. A systematic approach to the promotion will contribute significantly to the usefulness of the research project results, by increasing the added value, adding social touch, and will lead to a continuation of activities in this area even after the end of the project.