The colour theorems of Brooks and Gallai extended
Discrete Mathematics
Color-critical graphs on a fixed surface
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Introduction to Algorithms
Acyclic Homomorphisms and Circular Colorings of Digraphs
SIAM Journal on Discrete Mathematics
Circular colorings of edge-weighted graphs
Journal of Graph Theory
The circular chromatic number of a digraph
Journal of Graph Theory
Digraphs: Theory, Algorithms and Applications
Digraphs: Theory, Algorithms and Applications
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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.