P=NP

TL;DR

This paper explores the NP-complete graph 3-coloring problem for graphs with maximum degree 4 by linking it to semidefinite programming, establishing a new characterization of 3-colorability via optimization bounds.

Contribution

It introduces a novel semidefinite programming formulation R(G) that characterizes 3-colorability of graphs with degree at most 4, providing a new perspective on the problem.

Findings

G is 3-colorable iff the objective of R(G) is bounded.

When bounded, the minimum of R(G) is 0.

The paper establishes a direct link between graph coloring and semidefinite programming.

Abstract

This paper investigates an extremely classic NP-complete problem: How to determine if a graph G, where each vertex has a degree of at most 4, can be 3-colorable(The research in this paper focuses on graphs G that satisfy the condition where the degree of each vertex does not exceed 4. To conserve space, it is assumed throughout the paper that graph G meets this condition by default.). The author has meticulously observed the relationship between the coloring problem and semidefinite programming, and has creatively constructed the corresponding semidefinite programming problem R(G) for a given graph G. The construction method of R(G) refers to Theorem 1.1 in the paper. I have obtained and proven the conclusion: A graph G is 3-colorable if and only if the objective function of its corresponding optimization problem R(G) is bounded, and when the objective function is bounded, its minimum value is 0.