Graph Theory: Colourings, flows, and decompositions (GRACOL) Start date: Feb 1, 2013, End date: Jan 31, 2018 PROJECT  FINISHED

Graph theory is a relatively new branch of mathematics. Early sources of inspiration are Kirchhoff’s theory of electrical networks and the 4-color problem, both from the 19th century. In the 20th century graph theory was one of the most rapidly growing branches of mathematics with applications to theoretical computer science (design and analysis of algorithms), operations research (combinatorial optimization) and models in engineering and economics. The internet may be thought as a graph. There are also strong ties to geometry, topology, probability theory and logic.The main subjects in the project are graphs in the plane and on higher surfaces, graph decomposition, the Tutte polynomial and the graph flow conjectures, and also combinatorial problems arising from differential geometry. The project is centered around applying new approaches to some classical problems in graph theory, in particular problems in chromatic graph theory and flow theory. In some sense these problems have an algebraic unification in the Tutte polynomial of two variables. The Tutte polynomial has as special valuations (fixing one of the variables) the chromatic polynomial (introduced in 1912 by Birkhoff) and the flow polynomial. More recently, the Tutte polynomial has also become of interest in statistical mechanics.Among the specific problems to be investigated is Tutte’s 3-flow conjecture from the early 1970es, the problem if the flow polynomial can have arbitrarily large roots (motivated by Tutte’s 5-flow conjecture), the Merino-Welsh conjecture on the numbers of spanning trees, acyclic orientations and totally cyclic orientations, and Wegner’s conjecture from 1977 about squares of planar cubic graphs. We expect to get significant new insight (but not complete solutions) to the two notoriously hard flow conjectures of Tutte (both of which are also described in Wikipedia). We expect to almost solve the Merino-Welsh conjecture. We expect to completely solve the Wegner conjecture.