Minimal imperfect graphs: A simple approach


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.

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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.

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Gasparian, G.S. Minimal imperfect graphs: A simple approach. Combinatorica 16, 209–212 (1996).

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Mathematics Subject Classification (1991)

  • 05 C