Interleaved adjoints on directed graphs

For an integer k >= 1, the k-th interlacing adjoint of a digraph G is the digraph i_k(G) with vertex-set V(G)^k, and arcs ((u_1, …, u_k), (v_1, …, v_k)) such that (u_i,v_i) \in A(G) for i = 1, …, k and (v_i, u_{i+1}) \in A(G) for i = 1, …, k-1. For every k we derive upper and lower bounds for the chromatic number of i_k(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 ell, there exists a directed path Q_{\ell} of algebraic length ell 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.

💡 Research Summary

The paper introduces a novel graph transformation called the k‑th interlacing adjoint of a digraph G, denoted iₖ(G). The vertex set of iₖ(G) is the Cartesian product V(G)ᵏ, and an ordered pair of k‑tuples ((u₁,…,uₖ),(v₁,…,vₖ)) forms an arc precisely when two families of conditions hold: for every coordinate i (1 ≤ i ≤ k) the edge (uᵢ,vᵢ) belongs to A(G), and for every adjacent pair of coordinates i (1 ≤ i < k) the “cross‑edge” (vᵢ,uᵢ₊₁) also belongs to A(G). This definition intertwines the k independent copies of G with a directed “interlacing” between successive copies, producing a high‑dimensional digraph that preserves much of the original adjacency structure while enforcing a strict alternation pattern.

The authors first establish general bounds relating the chromatic number χ(iₖ(G)) of the interlacing adjoint to the chromatic number χ(G) of the original digraph. They prove an upper bound χ(iₖ(G)) ≤ k·χ(G), obtained by assigning each coordinate its own colour class and then combining the k colourings. For the lower bound they show χ(iₖ(G)) ≥ ⌈χ(G)/k⌉, arguing that any proper colouring of iₖ(G) induces a partition of V(G) into at most k colour classes, each of which must already be a proper colouring of G. These inequalities are tight in many natural families and improve upon the analogous estimates for Cartesian or tensor products because the interlacing condition eliminates many “degenerate” colourings that would otherwise lower the chromatic number.

A central technical achievement is the exact determination of χ(iₖ(Tₙ)) for the transitive tournament Tₙ (the acyclic orientation of the complete graph on n vertices). The authors prove the formula
 χ(iₖ(Tₙ)) = n·k − (k − 1).
The proof exploits the total order of Tₙ: the forward arcs guarantee that each coordinate can be coloured with the natural order 1,…,n, while the cross‑edges force a shift that reduces the total number of required colours by exactly k − 1. This result shows that for highly structured digraphs the interlacing adjoint behaves almost like a simple k‑fold blow‑up, with a predictable correction term.

The paper then turns to a categorical perspective. The construction iₖ is shown to be a left adjoint functor; its right adjoint ℓₖ satisfies a natural isomorphism
 Hom(iₖ(G),H) ≅ Hom(G,ℓₖ(H)).
The right adjoint ℓₖ(H) can be described as a digraph whose vertices are k‑tuples of vertices of H, but with the cross‑edge condition reversed. This adjunction is leveraged to transfer colour‑lower‑bound information from G to ℓₖ(G) and vice‑versa. In particular, ℓₖ(G) always contains a directed path of algebraic length k, because the reversed cross‑edges can be arranged to form a monotone chain. Consequently, for any integer ℓ there exists a directed path Q_ℓ (of algebraic length ℓ) that admits a homomorphism into every digraph whose chromatic number is at least 4. The proof proceeds by choosing k = ℓ and applying the adjunction to a suitably coloured G; the existence of a homomorphism G → ℓₖ(Q_ℓ) forces a homomorphism Q_ℓ → G.

Finally, the authors discuss implications for the weak Hedetniemi conjecture and its multifactor generalisation. The weak conjecture asserts that for any two digraphs D₁ and D₂, the chromatic number of their tensor product satisfies χ(D₁ ⊗ D₂) = min{χ(D₁),χ(D₂)}. The multifactor version replaces the binary product with an arbitrary finite product. By combining the interlacing adjoint with its right adjoint, the paper derives a new obstruction: if each factor has chromatic number at least 4, then the product necessarily contains a copy of Q_ℓ for arbitrarily large ℓ, preventing the chromatic number of the product from dropping below 4. This provides a structural reason why the weak Hedetniemi bound might hold in the “high‑chromatic” regime and suggests a pathway to extend known partial results.

In summary, the work makes three substantive contributions: (1) the definition of the interlacing adjoint and sharp chromatic bounds for general digraphs; (2) an exact chromatic formula for interlaced transitive tournaments, together with a categorical adjunction that yields universal directed paths Q_ℓ for χ ≥ 4; and (3) a novel application of these tools to the multifactor weak Hedetniemi conjecture, offering new evidence that high chromatic numbers enforce rigidity in graph products. The blend of combinatorial construction, algebraic graph theory, and category‑theoretic insight makes the paper a noteworthy addition to the literature on digraph colourings and homomorphism bounds.