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A ( c 1 , c 2 , … , c k ) -coloring of a graph G is a mapping φ : V ( G ) ↦ { 1 , 2 , … , k } such that for every i , 1 ≤ i ≤ k , G [ V i ] has maximum degree at most c i , where G [ V i ] denotes the subgraph induced by the vertices colored i . Borodin and Raspaud conjecture that every planar graph with neither 5 -cycles nor intersecting triangles is 3 -colorable. We prove in this paper that every planar graph with neither 5 -cycles nor intersecting triangles is (2, 0, 0)-colorable. [ABSTRACT FROM AUTHOR]

We define a map F with domain a certain subset of the set of light leaves (certain morphisms between Soergel bimodules introduced by the author in an earlier paper) and range the set of prime numbers. Using results of Soergel we prove the following property of F : if the image p = F ( l ) of some light leaf l under F is bigger than the Coxeter number of the corresponding Weyl group, then there is a counterexample to Lusztig's conjecture in characteristic p . We also introduce the “double leaves basis” which is an improvement of the light leaves basis that has already found interesting applications. In particular it forms a cellular basis of Soergel bimodules that allows us to produce an algorithm to find “the bad primes” for Lusztig's conjecture. [ABSTRACT FROM AUTHOR]