    A Constructive Formalization of the Weak Perfect Graph Theorem

The Perfect Graph Theorems are important results in graph theory describing the relationship between clique number ω(G) and chromatic number χ(G) of a graph G. A graph G is called perfect if χ(H)=ω(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 Lovász 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.

Authors

12/28/2018

Towards a constructive formalization of Perfect Graph Theorems

Interaction between clique number ω(G) and chromatic number χ(G) of a ...
09/03/2018

An Optimal χ-Bound for (P_6, diamond)-Free Graphs

Given two graphs H_1 and H_2, a graph G is (H_1,H_2)-free if it contains...
06/16/2021

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete ...
02/08/2020

Majority Voting and the Condorcet's Jury Theorem

There is a striking relationship between a three hundred years old Polit...
04/13/2019

Minimal Separators in Graphs

The Known Menger's theorem states that in a finite graph, the size of a ...
01/07/2018

Perfect graphs with polynomially computable kernels

In a directed graph, a kernel is a subset of vertices that is both stabl...
07/03/2018

Elusive extremal graphs

We study the uniqueness of optimal solutions to extremal graph theory pr...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.