(Claw, C_3)-free digraphs with unbounded dichromatic number

We construct orientations of rook graphs (whose underlying graphs are claw-free) that contain no directed $C_3$ but have unbounded dichromatic number. This disproves a conjecture of Aboulker, Charbit and Naserasr and improves a result of Carbonero, Koerts, Moore and Spirkl.

💡 Research Summary

The paper addresses a long‑standing conjecture in the theory of directed graphs concerning the relationship between forbidden induced subdigraphs and the dichromatic number (the minimum number of colours needed to colour the vertices so that each colour class induces an acyclic subdigraph). Aboulker, Charbit and Naserasr conjectured that for any “hero” H (a tournament with bounded dichromatic number) and any oriented star forest S, every oriented graph that contains neither H nor S as an induced subdigraph should have bounded dichromatic number. Earlier works produced counter‑examples for very specific choices of H and S, but the conjecture remained open for many natural families, in particular when S is a claw (the oriented star K₁,₃) and H is the directed triangle ∇C₃.

The authors construct a family of digraphs Dₙ, each obtained by orienting the edges of the N × N rook graph Rₙ (the line graph of the complete bipartite graph K_{N,N}). Rₙ is claw‑free, so any orientation of it automatically satisfies the “claw‑free” condition on the underlying undirected graph. The orientation rule is defined via binary expansions of the row and column indices: for two vertices sharing a row, say (a,b) and (a,d), let i be the least index where the binary digits of b and d differ. The edge is oriented from (a,b) to (a,d) exactly when the i‑th bit of b equals the i‑th bit of a; an analogous rule applies to vertices sharing a column. This rule guarantees that any three vertices lying in the same row (or column) are oriented consistently, which immediately implies that Dₙ contains no directed 3‑cycle (Proposition 1).

A crucial structural observation (Lemma 1) shows that whenever the horizontal and vertical distances between two opposite corners of a rectangle are equal (|a−c| = |b−d|), the four corner vertices (a,b), (a,d), (c,d), (c,b) form a directed 4‑cycle. The proof again relies on the binary‑bit rule and demonstrates that the orientation of each side of the rectangle follows a deterministic pattern, yielding a directed cycle of length four.

To prove that the dichromatic number of Dₙ is unbounded, the authors invoke the Gallai–Witt theorem (also known as the multidimensional van der Waerden theorem). This theorem asserts that for any fixed number of colours k there exists a sufficiently large N such that any k‑colouring of the integer grid