Gallai's Theorem for List Coloring of Digraphs

Abstract

A classical theorem of Gallai states that in every graph that is critical for k-colorings, the vertices of degree $k-1$ induce a tree-like graph whose blocks are either complete graphs or cycles of odd length. We provide a generalization to colorings and list colorings of digraphs, where some new phenomena arise. In particular, the problem of list coloring digraphs with the lists at each vertex v having $\min\{d^{+}(v),d^{-}(v)\}$ colors turns out to be NP-hard.