Interlacing adjoints on directed graphs

For an integer k ≥ 1, the k-th interlacing adjoint of a digraph G is the digraph ιk(G) with vertex-set V (G) k , and arcs ((u1, . . . , uk), (v1, . . . , vk)) such that (ui, vi) ∈ A(G) for i = 1, . . . , k and (vi, ui+1) ∈ A(G) for i = 1, . . . , k − 1. For every k we derive upper and lower bounds for the chromatic number of ιk(G) in terms of that of G. In particular, we find tight bounds on the chromatic number of interlacing adjoints of transitive tournaments. We use this result in conjunction with categorial properties of adjoint functors to derive the following consequence. For every integer l, there exists a directed path Ql of algebraic length l which admits homomorphisms into every directed graph of chromatic number at least 4. We discuss a possible impact of this approach on the multifactor version of the weak Hedetniemi conjecture.