A Critique of Deng's "P=NP"

TL;DR

This paper critically analyzes Deng's claimed polynomial-time solution for 3-coloring graphs with degree at most 4, identifying a key error that invalidates the proof that P equals NP.

Contribution

It provides a detailed critique of Deng's approach, highlighting a fundamental mistake involving subgraph concepts that undermines the claimed proof.

Findings

Identifies a conflation of subgraphs and induced subgraphs in Deng's proof

Shows that Deng's polynomial-time algorithm for 3-coloring is invalid

Concludes Deng's proof does not establish P=NP

Abstract

In this paper, we critically examine Deng's "P=NP" [Den24]. The paper claims that there is a polynomial-time algorithm that decides 3-coloring for graphs with vertices of degree at most 4, which is known to be an NP-complete problem. Deng presents a semidefinite program with an objective function that is unboundedly negative if the graph is not 3-colorable, and a minimum of 0 if the graph is 3-colorable. Through detailed analysis, we find that Deng conflates subgraphs with induced subgraphs, leading to a critical error which thereby invalidates Deng's proof that $\text{P}=\text{NP}$.