DICHROMATIC NUMBER AND FRACTIONAL CHROMATIC NUMBER
The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a directed cycle under this orientation. In 1979, Erdős and Neuma...
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Veröffentlicht in: | Forum of mathematics. Sigma 2016, Vol.4, Article e32 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | The dichromatic number of a graph
$G$
is the maximum integer
$k$
such that there exists an orientation of the edges of
$G$
such that for every partition of the vertices into fewer than
$k$
parts, at least one of the parts must contain a directed cycle under this orientation. In 1979, Erdős and Neumann-Lara conjectured that if the dichromatic number of a graph is bounded, so is its chromatic number. We make the first significant progress on this conjecture by proving a fractional version of the conjecture. While our result uses a stronger assumption about the fractional chromatic number, it also gives a much stronger conclusion: if the fractional chromatic number of a graph is at least
$t$
, then the fractional version of the dichromatic number of the graph is at least
${\textstyle \frac{1}{4}}t/\log _{2}(2et^{2})$
. This bound is best possible up to a small constant factor. Several related results of independent interest are given. |
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ISSN: | 2050-5094 2050-5094 |
DOI: | 10.1017/fms.2016.28 |