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Complete Acyclic Colorings.
TL;DR: Two parameters that arise from the dichromatic number and the vertex-arboricity, the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph, are studied.
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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.
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Citations
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On sensitivity in bipartite Cayley graphs.
TL;DR: It is believed that Coxeter groups are a suitable generalization of the hypercube with respect to Huang's question and three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree $1$ on more than half the vertices are presented.
References
The chromatic number of random graphs
TL;DR: For a fixed probability p, 0 < p < 1, almost every random graph Gn,p has chromatic number σ 2 + o(1) + σ σ 1/(1 - p)) σ n{log n} as mentioned in this paper.
448
Chromatic number of random graphs
Béla Bollobás
- 01 Jan 1988
TL;DR: It is shown that for a fixed probabilityp, 0
322
On independent circuits contained in a graph
Paul Erdös,L. Pósa +1 more
TL;DR: In this article, a family of circuits of a graph G is said to be independent if no two of the circuits have a common vertex; it is called edge-independent if none two of them have an edge in common.
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Packing directed circuits
TL;DR: It is proved that for every digraphG, either G hasn vertex-disjoint directed circuits, or G can be made acyclic by deleting at mostt vertices.
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