In the 1960s, Minty, Gallai, and Roy proved that k-colorability of graphs has equivalent conditions in terms of the existence of orientations containing no cycles resp. paths with some orientation patterns. We give a common generalization of those classic results, providing new (necessary and sufficient) conditions for a graph to be k-chromatic. We also prove that if an orientation with those properties is available, or cycles of given lengths are excluded, then a proper coloring with a small number of colors can be found by a fast-linear or polynomial-algorithm. The basic idea of the proofs is to introduce directed and weighted variants of depth-first-search trees. Several related problems are raised.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics