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The dichromaticÌýnumberÌýof a directed graph D is the minimum integer k such that the vertex set of D can be partitioned into k acyclic subdigraphs. It is easy to see that the chromaticÌýnumberÌýof a graph G is the dichromaticÌýnumberÌýof the digraph obtained from G by replacing each edge with a digon (two anti-parallel arcs). Based on this simple observation, many theorems concerning the chromaticÌýnumberÌýof undirected graphs have been generalized to digraphs via dichromaticÌýnumber. However,ÌýnoÌýconcept analogous to cliqueÌýnumberÌýfor digraphs has been available. The purpose of this presentation is to explore such a concept and its relationship with the dichromaticÌý number, mirroring the relationship between theÌýcliqueÌýnumberÌýand the chromaticÌýnumber in undirected graphs. We will focus, in particular, on studying the notion of χ-boundedness.
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