Abstract
In this paper, we study colorings of k-partite sparse digraphs. The chromatic number of a graph G is the smallest integer k such that the vertices of G can be colored with k colors with the property that each color class is an independent set. The dichromatic number of a digraph D is the minimum k such that the vertices of D can be colored with k colors with each color class inducing an acyclic subdigraph. This coloring invariant shares many similarities with the graph chromatic number and can be thought of as its analogous digraph generalization.Our main result in this short note shows that there exist sparse k-partite digraphs which have dichromatic number k. This, in particular, not only implies that there exist graphs with equal chromatic and dichromatic number, but that they can be taken to be somewhat sparse.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.