Diperfect graphs

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 |μiS|=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.

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Dedicated to Tibor Gallai on his seventieth birthday

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Berge, C. Diperfect graphs. Combinatorica 2, 213–222 (1982). https://doi.org/10.1007/BF02579229

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AMS subject classification (1980)

  • 05 C 20
  • 05 C 15