Diperfect graphs |
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Authors: | Claude Berge |
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Institution: | (1) Centre de Mathématique Sociale, 54 Boulevard Raspail, 75270 Paris Cedex 06, France |
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Abstract: | Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ1, μ2, ..., μk such tht |μi ∩S|=1 for alli.
Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic
number; here again, we do not know if there exists an optimal coloring (S
1,S
2, ...,S
k) and a path μ such that |μ ∩S
i|=1 for alli.
In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable
setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal
coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the
perfect graph conjecture.
Dedicated to Tibor Gallai on his seventieth birthday |
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Keywords: | 05 C 20 05 C 15 |
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