Vertex 2-coloring without monochromatic cycles of fixed size is NP-complete

In this paper we study a problem of vertex two-coloring of an undirected graph such that there is no monochromatic cycle of the given length. We show that this problem is hard to solve. We give a proof by presenting a reduction from the variation of satisfiability (SAT) problem. We show the nice properties of coloring cliques with two colors which plays pivotal role in the reduction construction.