ICLA 2019: Logic and Its Applications pp 183-194

# Towards a Constructive Formalization of Perfect Graph Theorems

• Abhishek Kr Singh
• Raja Natarajan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11600)

## Abstract

Interaction between clique number $$\omega (G)$$ and chromatic number $$\chi (G)$$ of a graph is a well studied topic in graph theory. Perfect Graph Theorems are probably the most important results in this direction. Graph G is called perfect if $$\chi (H)=\omega (H)$$ for every induced subgraph H of G. The Strong Perfect Graph Theorem (SPGT) states that a graph is perfect if and only if it does not contain an odd hole (or an odd anti-hole) as its induced subgraph. The Weak Perfect Graph Theorem (WPGT) states that a graph is perfect if and only if its complement is perfect. In this paper, we present a formal framework for verifying these results. We model finite simple graphs in the constructive type theory of Coq Proof Assistant without adding any axiom to it. Finally, we use this framework to present a constructive proof of the Lovász Replication Lemma, which is the central idea in the proof of Weak Perfect Graph Theorem.

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