Interleaved adjoints of directed graphs

For an integer k ≥ 1 , the k th interleaved adjoint of a digraph G is the digraph ι k ( G ) with vertex-set V ( G ) k , and arcs ( ( u 1 , … , u k ) , ( v 1 , … , v k ) ) such that ( u i , v i ) ∈ A ( G ) for i = 1 , … , k and ( v i , u i + 1 ) ∈ A ( G ) for i = 1 , … , k − 1 . For every k , we derive upper and lower bounds for the chromatic number of ι k ( G ) in terms of that of G . In the case where G is a transitive tournament, the exact value of the chromatic number of ι k ( G ) has been determined by [H.G. Yeh, X. Zhu, Resource-sharing system scheduling and circular chromatic number, Theoret. Comput. Sci. 332 (2005) 447–460]. We use the latter result in conjunction with categorical properties of adjoint functors to derive the following consequence. For every integer ℓ , there exists a directed path Q ℓ 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.