Conjecture   If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.

Conjecture   Every digraph in which each vertex has outdegree at least contains directed cycles such that meets in at most one vertex, .

Problem   Does there exist a polynomial time algorithm which takes as input a graph and for every vertex a subset of , and decides if there exists a partition of into so that only if and so that are independent, is a clique, and there are no edges between and ?