## Final Report Summary - GRACOL (Graph Theory: Colourings, flows, and decompositions.)

The GRACOL-research was in mathematics, specifically graph theory, and focused on colorings, flows and decompositions. Graph coloring started with the fundamental 4-color problem from the 19th century and has become important in computer science, engineering sciences, and also operations research as many scheduling problems can be expressed as graph coloring problems. The subject has a vast number of challenging open problems some of which have been attacked very successfully by a truly global team brought together in GRACOL.

Thus we obtained a complete solution of the Wegner conjecture from 1977 saying that the square of every cubic planar graph is 7-colorable. We also obtained a complete solution of the Bárat-Thomassen conjecture from 2006 which is a decomposition statement for each finite tree T. And we obtained partial progress on a number of long-standing open problems, in particular the (so far) best result on the Merino-Welsh conjecture from 1999. The classical and still wide open Birkhoff –Lewis conjecture from 1946 is about roots of chromatic polynomials of planar graphs in the interval between 4 and 5 (and that conjecture was again motivated by the classical 4-color problem). We solved almost completely the analogous problem for the other problematic interval (for this problem), namely the interval between 3 and 4.

GRACOL also has ties to theoretical computer science. The research was carried out in the section AlgoLoG (Algorithms, Logic and Graphs) at the Department of Applied Mathematics and Computer Science at DTU. The importance of graph theory to algorithms has been a key factor in the project. For example, a characterization has been obtained of the graphs having a strongly 2-connected orientation (an open problem formulated in the 1995 by A. Frank). This leads to a fast algorithm for deciding if a graph admits such an orientation. (For strongly 3-connected orientations the problem is NP-complete.) Also an algorithm has been described for verifying the Merino-Welsh conjecture for graphs of any fixed path-width, and this has been used to actually verify the conjecture for graphs of small path-width.

Thus we obtained a complete solution of the Wegner conjecture from 1977 saying that the square of every cubic planar graph is 7-colorable. We also obtained a complete solution of the Bárat-Thomassen conjecture from 2006 which is a decomposition statement for each finite tree T. And we obtained partial progress on a number of long-standing open problems, in particular the (so far) best result on the Merino-Welsh conjecture from 1999. The classical and still wide open Birkhoff –Lewis conjecture from 1946 is about roots of chromatic polynomials of planar graphs in the interval between 4 and 5 (and that conjecture was again motivated by the classical 4-color problem). We solved almost completely the analogous problem for the other problematic interval (for this problem), namely the interval between 3 and 4.

GRACOL also has ties to theoretical computer science. The research was carried out in the section AlgoLoG (Algorithms, Logic and Graphs) at the Department of Applied Mathematics and Computer Science at DTU. The importance of graph theory to algorithms has been a key factor in the project. For example, a characterization has been obtained of the graphs having a strongly 2-connected orientation (an open problem formulated in the 1995 by A. Frank). This leads to a fast algorithm for deciding if a graph admits such an orientation. (For strongly 3-connected orientations the problem is NP-complete.) Also an algorithm has been described for verifying the Merino-Welsh conjecture for graphs of any fixed path-width, and this has been used to actually verify the conjecture for graphs of small path-width.