Let
k and
r be two integers with
k≥
2 and
k≥
r≥
1. We show that
- if a strongly
connected digraph D contains no directed cycle of length 1 modulo k, then D is
k-colorable;
- if a digraph D contains no directed cycle of length r modulo k, then D can be vertex-colored
with k colors so that each color class induces an acyclic subdigraph.
In addition to strengthening several classical graph coloring results (by Bondy, Erdős and Hajnal, Gyárfás, etc), our results give an affirmative
answer to a question of Tuza in 1992 and imply a strong form of a conjecture of
Diwan, Kenkre and Vishwanathan: If graph
G contains no cycle of length
r modulo
k, then
G is
k-colorable if
r≠
2 and
(k+1)-colorable otherwise. In this talk, we will also discuss other related results and mention sereval open problems (both graph theoretic and algorithmic). Most of the results presented are joint with Zhibin Chen and Wenan Zang.
The ACO Seminar (2014–2015)