Head of program P1-0297: prof. dr. Sandi Klavžar

Duration: 01/01/2022 - 12/31/2027

Abstract: The research programme of our group is to remain at the forefront of research in the field of graph theory. We will continue researching central, standard areas of graph theory. In addition to that, we intend to be one step ahead of the competition and explore and introduce new concepts, techniques, and applications. Among the areas that we especially plan to develop are domination theory, metric graph theory, graph colorings, chemical graph theory, graph products, hamiltonicity, topological and geometric graph theory, and applications of these areas. A brief summary of some of the problems and topics that we will investigate follows. In domination theory we will investigate the number of optimal sets for domination invariants, the problem of finding the graphs with unique optimal sets, different domination games, and Grundy invariants. In metric graph theory we will focus on the variety of metrically defined subclasses of hypercubes, the Hausdorff distance, the Wiener index of digraphs, and relations of metric graph theory with oriented matroids. We will especially investigate daisy cubes and daisy graphs of rooted graphs, Pell graphs, cube complements of graphs, and a connection between statistical learning and geometric properties of subgraphs of hypercubes. The main themes of graph colorings will be packing and S-packing colorings, and different distance colorings. In chemical graph theory we plan to develop a generalized approach to determine the extremal structures for some degree-based topological indices, investigate resonance graphs, and determine the best performing model for the prediction of physico-chemical properties of unsaturated hydrocarbons. For graph products we plan to study some specific problems and consider in depth properties of the direct-co-direct product and the modular product. For hamiltonicity we will investigate the problem whether Thomassen's result can be improved for (list) colorings of Eulerian triangulations, investigate the problem whether Chvatal's conjecture holds for bipartite graphs and investigate whether all 1-tough bipartite graphs are prism-hamiltonian. In the area of the topological and geometric graph theory we will study the structure of crossing critical graphs, genus and crossings of sparse, dense, and random objects, graph and subgraph isomorphism of geometrically represented graphs, and the use of algebraic techniques in geometric and topological graphs. Along with many of the listed topics, algorithmic aspects will also be investigated.

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