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P=NP

Zikang Deng

TL;DR

The paper addresses deciding 3-colorability for graphs with maximum degree $4$ by formulating a semidefinite program $R(G)$ whose objective $f(G)$ is bounded and attains a minimum of $0$ iff $G$ is 3-colorable, thereby linking graph coloring to convex optimization. It develops a dual copositive/completely positive programming framework with the notions of D-graphs to analyze feasibility and unboundedness, using duality to connect 3-colorability with the properties of the SDP and its duals. A key claim is that introducing a numeric constraint $f(G)\ge -100$ yields a polynomial-time solvable formulation, which the author argues would imply $P=NP$ for graphs with degree at most $4$. The work combines SDP theory, copositive/CP formulations, and graph-theoretic structures to establish a novel theoretical bridge between combinatorial coloring problems and convex optimization, with significant implications for complexity theory and optimization-based approaches.

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.

P=NP

TL;DR

The paper addresses deciding 3-colorability for graphs with maximum degree by formulating a semidefinite program whose objective is bounded and attains a minimum of iff is 3-colorable, thereby linking graph coloring to convex optimization. It develops a dual copositive/completely positive programming framework with the notions of D-graphs to analyze feasibility and unboundedness, using duality to connect 3-colorability with the properties of the SDP and its duals. A key claim is that introducing a numeric constraint yields a polynomial-time solvable formulation, which the author argues would imply for graphs with degree at most . The work combines SDP theory, copositive/CP formulations, and graph-theoretic structures to establish a novel theoretical bridge between combinatorial coloring problems and convex optimization, with significant implications for complexity theory and optimization-based approaches.

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.
Paper Structure (7 sections, 9 theorems, 81 equations)

This paper contains 7 sections, 9 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Example
  3. Some important theorems
  4. Proof of Theorem 3.2 and Theorem 3.3
  5. Proof of Theorem 1.1 and Theorem 3.1
  6. P=NP
  7. References

Key Result

Theorem 1.1

For a given graph $G=(V, E)$ of order $n$(The degree of each vertex in graph G is at most 4.) , the following semi-definite program is constructed((1)-(17)): $min_{d_*,p_*}$ $s.t.$ If vertices $v_i$ and $v_j$ in graph $G$ are adjacent($i \neq j$), then ((6)): If vertices $v_i$ and $v_j$ in graph $G$ are not adjacent($i \neq j$), then((7)): If vertices $v_i$ and $v_j$ in graph $G$ are adjacent($

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5