Properties of random 2-primitive digraphs and matrices

A digraph in which each arc is colored either red or blue is 2-primitive if there are integers r and b such that there is a walk consisting of exactly r red arcs and b blue arcs, in any order, between any 2 vertices. All balanced 2-colorings of complete digraphs are 2-primitive with 2-exponent at most (2, 2). For every primitive digraph there is a 2-colored balancing that is 2-primitive.; All primitive tournaments have a 2-primitive 2-coloring with nearly equal numbers of red and blue arcs. The probability that a randomly 2-colored primitive tournament is 2-primitive is asymptotically 1 as the number of vertices goes to infinity. However, k-regular digraphs are asymptotically almost never 2-primitive.; Every primitive digraph containing an even number of edge-independent cycles whose set of cycle lengths is relatively prime has a nontrivial 2-coloring that is 2-primitive.