Minimal imperfect graphs: A simple approach

Abstract

We provide a very short proof of the following theorem of Lovász, and of its consequences:A graph is perfect if and only if in every induced subgraph the number of vertices does not exceed the product of the stability and clique numbers of the subgraph.

This proof is conceptually new: it does not use the “replication” operation, or any kind of polyhedral argument; the arguments resemble more the well-known ways of deducing structural properties of minimal imperfect graphs. However, the known proofs of these structural properties use Lovász's result, whereas the present work leads to a proof of Lovász's result itself, and actually to a slight sharpening.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    R. G. Bland, H.-C. Huang, andL. E. Trotter Jr.: Graphical properties related to minimal imperfection,Discrete Math,27 (1979), 11–22.

    Article  Google Scholar 

  2. [2]

    C. Berge: Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind,Wiss. Z. Martin Luther King Univ. Halle-Wittenberg, 1961,114.

  3. [3]

    V. Chvátal: On certain polytopes associated with graphs,J. of Combin. Theory Ser. B,18 (1975), 138–154.

    Article  Google Scholar 

  4. [4]

    D. R. Fulkerson: The perfect graph conjecture and the pluperfect graph theorem, in:Proceedings of the Second Chapel Hill Conference on Combinatorial Mathematics and its applications, (R. C. Bose et al. eds),1, (1970) 171–175.

  5. [5]

    L. Lovász: Normal Hypergraphs and the Perfect Graph Conjecture,Discrete Mathematics,2 (1972), 253–268.

    Article  Google Scholar 

  6. [6]

    L. Lovász: A characterization of perfect graphs,J. of Combin. Theory,13 (1972), 95–98.

    Article  Google Scholar 

  7. [7]

    L. Lovász: Perfect Graphs, in:More Selected Topics on Perfect Graph Theory, L. M. Beineke and R. L. Wilson eds, Academic Press, London, NY, 55–87 (1972).

    Google Scholar 

  8. [8]

    M. Padberg: Perfect zero-one matrices,Math. Programming,6 (1974), 180–196.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

The author gratefully acknowledges financial support from “Ministère de la Recherche et de la Technologie” (the French Ministry of Research and Technology, MRT) which sponsored his three month visit to Laboratory ARTEMIS of Grenoble University which motivated him to do this work.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gasparian, G.S. Minimal imperfect graphs: A simple approach. Combinatorica 16, 209–212 (1996). https://doi.org/10.1007/BF01844846

Download citation

Mathematics Subject Classification (1991)

  • 05 C