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New Lower Bounds for C4-Free Subgraphs of the Hypercubes Q6, Q7, and Q8: Constructions, Structure, and Computational Method

Minamo Minamoto

Abstract

We establish new lower bounds ex(Q_7,C_4)>=304 and ex(Q_8,C_4)>=680 for the maximum number of edges in a C_4-free subgraph of the 7- and 8-dimensional hypercubes, and give a modern computational reproduction of ex(Q_6,C_4)=132. All bounds are witnessed by explicit constructions certified by exhaustive enumeration of all four-cycles (240 for Q_6, 672 for Q_7, 1792 for Q_8). For Q_7 we identify 19866 distinct C_4-free subgraphs on 304 edges and classify them into exactly 20 structural types via their dimension profiles. All Q_7 solutions share a rigid structural core: degree sequence {4^32,5^96}, spectral radius lambda_1 approximately 4.787, and local maximality. For Q_8 we analyse the 680-edge construction and the 681-edge barrier: every non-edge creates at least one C_4, and 1076 independent searches at 681 edges never achieved zero violations. The constructions are found by a two-phase simulated annealing algorithm with Aut(Q_n)-based diversification. For Q_6 we provide an ILP-based proof that ex(Q_6,C_4)<=132. Edge lists, ILP files, and source code are publicly available at https://github.com/minamominamoto/c4free-hypercube

New Lower Bounds for C4-Free Subgraphs of the Hypercubes Q6, Q7, and Q8: Constructions, Structure, and Computational Method

Abstract

We establish new lower bounds ex(Q_7,C_4)>=304 and ex(Q_8,C_4)>=680 for the maximum number of edges in a C_4-free subgraph of the 7- and 8-dimensional hypercubes, and give a modern computational reproduction of ex(Q_6,C_4)=132. All bounds are witnessed by explicit constructions certified by exhaustive enumeration of all four-cycles (240 for Q_6, 672 for Q_7, 1792 for Q_8). For Q_7 we identify 19866 distinct C_4-free subgraphs on 304 edges and classify them into exactly 20 structural types via their dimension profiles. All Q_7 solutions share a rigid structural core: degree sequence {4^32,5^96}, spectral radius lambda_1 approximately 4.787, and local maximality. For Q_8 we analyse the 680-edge construction and the 681-edge barrier: every non-edge creates at least one C_4, and 1076 independent searches at 681 edges never achieved zero violations. The constructions are found by a two-phase simulated annealing algorithm with Aut(Q_n)-based diversification. For Q_6 we provide an ILP-based proof that ex(Q_6,C_4)<=132. Edge lists, ILP files, and source code are publicly available at https://github.com/minamominamoto/c4free-hypercube

Paper Structure

This paper contains 17 sections, 7 theorems, 4 equations, 3 tables.

Table of Contents

  1. Introduction
  2. $C_4$ Enumeration and Verification
  3. The $Q_7$ Construction
  4. The $Q_8$ Construction
  5. The 681-edge barrier
  6. Structural Classification of $Q_7$ Solutions
  7. Dimension profiles and the twenty types
  8. Shared structural properties
  9. Solution landscape
  10. Computational Method
  11. Two-phase simulated annealing
  12. Comparison with known lower bounds
  13. Regularity trend across dimensions
  14. Computational Proof that $\mathrm{ex}(Q_6,C_4)=132$
  15. Lower bound: explicit construction
  16. ...and 2 more sections

Key Result

Theorem 1

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 2: $C_4$ enumeration in $Q_n$
  • Proposition 3
  • Proposition 4
  • Remark 5
  • Conjecture 6
  • Definition 7: Dimension profile
  • Theorem 8
  • Proposition 9
  • Conjecture 10
  • ...and 1 more