A Constructive Formalization of the Weak Perfect Graph Theorem

Abhishek Kr Singh; Raja Natarajan

doi:10.1145/3372885.3373819

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arXiv · 1912.02211

A Constructive Formalization of the Weak Perfect Graph Theorem

Abhishek Kr Singh·Raja Natarajan

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Abstract

The Perfect Graph Theorems are important results in graph theory describing the relationship between clique number $\omega(G) $ and chromatic number $\chi(G) $ of a graph $G$. A graph $G$ is called \emph{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 working with finite simple graphs. We model finite simple graphs in the Coq Proof Assistant by representing its vertices as a finite set over a countably infinite domain. We argue that this approach provides a formal framework in which it is convenient to work with different types of graph constructions (or expansions) involved in the proof of the Lov\'{a}sz Replication Lemma (LRL), which is also the key result used in the proof of Weak Perfect Graph Theorem. Finally, we use this setting to develop a constructive formalization of the Weak Perfect Graph Theorem.

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BibTeX RIS

Abhishek Kr Singh, Raja Natarajan. 2019-12-04. A Constructive Formalization of the Weak Perfect Graph Theorem. https://doi.org/10.1145/3372885.3373819

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