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

• The paper establishes improved lower bounds: at least 304 edges for Q7 and 680 edges for Q8 in C4-free subgraphs.
• It employs a two-phase simulated annealing algorithm and exhaustive 4-cycle enumeration to certify subgraph optimality.
• The computational approach reveals a rich structure of locally maximal configurations, supporting conjectures on the bounds' tightness.

New Lower Bounds for $C_4$-Free Subgraphs of Hypercubes $Q_7$ and $Q_8$

Problem Setting and Prior Bounds

This work addresses the extremal function $ex(Q_n, C_4)$: the maximum size of a subgraph of the $n$-dimensional hypercube $Q_n$ that is $C_4$-free, where $C_4$ denotes the 4-cycle. The determination of $ex(Q_n, C_4)$ is a classical open question in extremal graph theory, initially posed by Erdős. The hypercube $Q_n$ has $Q_7$0 vertices and $Q_7$1 edges, and the $Q_7$2-free constraint is highly nontrivial due to the abundance of 4-cycles in high-dimensional hypercubes.

Previous significant results include:

• Lower bounds from Brass, Harborth, and Nienborg (BHN), yielding $Q_7$3 for $Q_7$4, with a weaker variant for general $Q_7$5.
• An upper bound of $Q_7$6 via flag algebra methods, due to Balogh et al.

Exact values are known for $Q_7$7, while $Q_7$8 remained open.

Main Contributions

Explicit lower bounds are improved for $Q_7$9 and $Q_8$0:

• $Q_8$1
• $Q_8$2

These improvements are achieved via computational constructions and outperform previous estimates: the prior BHN method gave approximately $Q_8$3 and $Q_8$4 for $Q_8$5 and $Q_8$6, respectively. Both cases exhibit surplus: $Q_8$7 edges ($Q_8$8) for $Q_8$9 and $ex(Q_n, C_4)$0 edges ($ex(Q_n, C_4)$1) for $ex(Q_n, C_4)$2.

Methodology and Computational Approach

The study employs a two-phase simulated annealing algorithm:

1. Penalty Simulated Annealing: Minimizes a penalized objective $ex(Q_n, C_4)$3, with $ex(Q_n, C_4)$4 the number of $ex(Q_n, C_4)$5 violations, across millions of steps at varying temperatures and penalty weights.
2. Swap Annealing: Fixes the edge count and attempts edge swaps to reduce $ex(Q_n, C_4)$6 violations, at higher computational intensity for supersaturated candidates.

Both phases utilize randomization via hypercube automorphisms to diversify search trajectories. Certificates of $ex(Q_n, C_4)$7-freeness are obtained by exhaustive enumeration of all 4-cycles in $ex(Q_n, C_4)$8—guaranteed due to the explicit combinatorial structure of $ex(Q_n, C_4)$9—and algorithmic checks per Lemma 2 of the paper.

Structural Properties of Constructions

For $n$0, the optimal constructions all exhibit:

• Degree sequence: 32 vertices of degree 4, 96 of degree 5.
• Bipartiteness: Largest and smallest eigenvalues are equal in magnitude and opposite in sign, as expected.
• Trace condition: The trace of $n$1 equals the sum $n$2, reflecting absence of 4-cycles.
• Local maximality: Every non-edge addition induces a $n$3.

A striking computational finding is the abundance of such locally maximal structures: $n$4 pairwise edge-distinct $n$5-free subgraphs with 304 edges were discovered for $n$6, with Hamming distances between solutions ranging widely, indicating a rich and nontrivial solution landscape.

For $n$7, the degree sequence found is $n$8. The exact BHN-type formula does not apply as $n$9 is non-integral power of $Q_n$0, but the edge-count advantage over the estimate is clear.

Strong and Conjectured Results

Multiple lines of empirical evidence strongly support the tightness of the new lower bounds:

• Persistent failure to reach 305 edges in $Q_n$1-free subgraphs of $Q_n$2, across $Q_n$3 trials, with minimum violation (number of $Q_n$4s) never reaching zero.
• For $Q_n$5, attempts at $Q_n$6-edge graphs always resulted in at least one $Q_n$7, over $Q_n$8 trials.

Based on this evidence, the paper conjectures that $Q_n$9 and that $C_4$0 is optimal or nearly optimal for $C_4$1.

Theoretical and Practical Implications

The techniques and results offer direct numerical progress on small hypercube extremal functions, advance computational methods for extremal combinatorics, and illustrate the intricacies of local optimality and structure in forbidden subgraph problems. The explicit enumerative and verification algorithm demonstrates a scalable paradigm for certifying forbidden subgraph properties in exponentially large, high-regularity graphs.

Implications extend to theoretical questions about the uniqueness and diversity of extremal configurations, the applicability of automated proof techniques (e.g., flag algebras) to small $C_4$2, and the role of heuristic search algorithms in combinatorial optimization. The success of exhaustive and randomized search in revealing the diversity of extremal subgraph configurations is of particular practical interest for designing and analyzing large-scale combinatorial structures subject to local constraints.

Future directions include:

• Rigorous proof of the tightness of these lower bounds
• Extension to $C_4$3 with $C_4$4
• Analysis of possible improvements via advanced algebraic or probabilistic techniques, such as flag algebras

Conclusion

This work establishes improved explicit lower bounds for the number of edges in $C_4$5-free subgraphs of $C_4$6 and $C_4$7, constructs and certifies a large number of locally maximal solutions, and presents strong computational evidence for precise extremal values at these dimensions. The research exemplifies the efficacy of computational search and combinatorial enumeration methods in extremal graph theory and opens avenues for both theoretical and computational advances in hypercube subgraph problems (2603.29127).