Symmetric extension of the red–blue kernel theorem

Establish whether the red–blue arc conditions in Theorem 1 admit a more general formulation with symmetric implications while preserving the kernel-existence conclusion.

Background

Theorem 1 generalizes the Sands–Sauer–Woodrow kernel-existence theorem by imposing two asymmetric implications on directed paths of blue and red arcs. The authors note that the asymmetry is essential to their proof but leave unresolved whether a corresponding theorem can be formulated with symmetric conditions.

References

We emphasize that the conditions~\ref{item:b} and~\ref{item:r} are not symmetric. We do not know whether a more general statement with symmetric conditions is possible.

— Revisiting classical results on kernels in digraphs  (2502.02482 - Langlois et al., 4 Feb 2025) in Section 1, subsection “Digraphs with red and blue arcs,” immediately after Theorem 1