Kernel-perfect directed graph

A digraph kernel is an independent set $K$ of vertices such that each vertex outside $K$ has a directed edge from itself to a vertex in $K$. A directed graph is kernel-perfect if every induced subgraph has a digraph kernel. Preference orientations of the line graph of a bipartite graph are kernel-perfect: a stable matching in a bipartite graph is independent in the line graph, and stability supplies the required outgoing edge at every unmatched edge.

In a kernel-perfect directed graph, lists of size at least $d^{+}(v)+1$ admit a proper graph coloring. Choose a color $a$ and a digraph kernel in the subgraph on vertices whose lists contain $a$. Color that kernel $a$, delete it, and remove $a$ from the remaining lists. Every vertex losing a color also loses an outgoing neighbour; hence the inequality persists. Induction on the vertex count finishes the proof.