Abstract
We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.
Highlights
All digraphs and graphs in this paper are considered loopless
A complete coloring of a graph is a proper vertex coloring such that the identification of any two colors produces a monochromatic edge
There has been a substantial amount of research on the achromatic number since its introduction in [9], we refer to [11] and [5] for survey articles on this topic
Summary
All digraphs and graphs in this paper are considered loopless. For digraphs, we allow parallel and anti-parallel edges, graphs may have multiple edges. We investigate complete colorings corresponding to the above two coloring parameters, resulting in the adichromatic number of directed graphs and the a-vertex arboricity of undirected graphs. The a-vertex arboricity ava(G) of an undirected graph G is the largest number of colors that can be used such that every color induces a forest but in the merge of any two color classes there is a cycle. Such a coloring will be referred to as a complete (arboreal) coloring of G.
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