The exact generalized Turán number for \(C_6\) in \(C_8\)-free graphs

Abstract: For graphs $F$ and $H$, let $\ex(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex $H$-free graph. Gerbner, Győri, Methuku and Vizer proved that $\ex(n,C_6,C_8)=Θ(n<sup>3)$ and predicted that the unrestricted problem should have the same first-order asymptotics as the bipartite one. We determine the exact value for all sufficiently large $n$, showing that [ \ex(n,C_6,C_8)=6\binom{n-3}{3}+12(n-5). ] Moreover, the unique extremal graph is $K_3\vee (K_2\cup I_{n-5})$. The main new ingredient is a codegree decomposition for $C_8$-free graphs: a packing lemma for triangles in the linear-codegree graph recovers an almost spanning common neighborhood, and a defect-absorption argument upgrades this stability to the exact extremal graph.

• The paper establishes that ex(n, C6, C8) = 6⋅binom(n-3,3) + 12(n-5) for all sufficiently large n.
• It introduces innovative techniques such as codegree decomposition, fat triangle packing, and defect absorption to control subgraph counts.
• The study confirms the unique extremal graph as K3 ∨ (K2 ∪ I_{n-5}), setting a benchmark for generalized Turán problems.

Exact Generalized Turán Number for $C_6$ in $C_8$-Free Graphs

Problem Context and Motivation

This paper addresses a longstanding open problem on generalized Turán numbers, focusing on $ex(n,C_6,C_8)$—the maximal number of (not necessarily induced) copies of the 6-cycle $C_6$ in an $n$-vertex $C_8$-free graph. While the order of growth $ex(n,C_6,C_8) = \Theta(n^3)$ was previously established, the precise leading constant, extremal configuration, and uniqueness for large $n$ had remained unresolved.

Generalized Turán problems for cycles, in particular for pairs $(C_{2\ell}, C_{2k})$, exhibit rich structure, especially in the sparse regime where even cycles are forbidden. In this setting, edge density fails to capture the local extremal behavior, and finer combinatorial parameters—such as codegrees and the structure of common neighborhoods—become central to the analysis.

Main Results

The authors prove the exact result: $ex(n, C_6, C_8) = 6\binom{n-3}{3} + 12(n-5)$ for all sufficiently large $C_8$0, and further establish that the unique extremal graph is the join $C_8$1, where $C_8$2 denotes the $C_8$3-vertex edgeless graph.

This resolves in full the precise asymptotics and extremal structure for $C_8$4, confirming and strengthening prior conjectures and first-order estimates in both bipartite and general settings.

Technical Approach

The proof leverages innovations in codegree decomposition and absorption methods:

• Fat Triangle Packing: The argument begins by introducing a codegree-threshold subgraph $C_8$5 (for large $C_8$6), whose triangles represent dense local configurations. A crucial packing lemma bounds the number of such triangles and links them to almost-spanning common neighborhoods, enabling the capture of the dominant contribution to $C_8$7 counts.
• Charging and Decomposition: Every $C_8$8 copy is "charged" to one of its alternating triples, based on average codegree. An advanced combinatorial analysis, separating cases depending on the number of high-codegree pairs ("fatness"), permits fine-grained control over the enumeration and enables tight error bounds.
• Defect Absorption and Stability: A stability lemma ensures that extremal graphs must have an almost-spanning triple with a common neighborhood, sharply restricting possible configurations. An intricate absorption argument then eliminates residual exceptional vertices, showing that any small defect reduces the $C_8$9 count below the extremal value, culminating in the uniqueness claim.

Key Numerical Claims

• For all large $ex(n,C_6,C_8)$0,

$ex(n,C_6,C_8)$1

• The extremal graph is uniquely $ex(n,C_6,C_8)$2.
• The presence of any vertices not in the prescribed core structure forces a strict decrease in the $ex(n,C_6,C_8)$3 count, validating the stability and absorption arguments.

Implications and Future Directions

The result provides the definitive answer for generalized extremal functions involving $ex(n,C_6,C_8)$4 in $ex(n,C_6,C_8)$5-free graphs. These methods, particularly the use of codegree decomposition, packing lemmas for fat triangles, and the defect-absorption procedure, demonstrate powerful techniques for handling generalized Turán problems in sparse settings.

From a theoretical standpoint, the precise determination of $ex(n,C_6,C_8)$6 and extremal graphs advances the understanding of how forbidden even cycles constrain higher-order subgraph counts, offering parallels to classical stability and extremality in dense graph theory, but in a fundamentally sparser context.

The techniques introduced here are likely to be adaptable to broader classes of subgraph-counting extremal problems, including those involving longer cycles or other bipartite substructures. The results suggest new directions in the classification of extremal graphs by forbidden even cycles, and may inform analogous questions in hypergraph extremal theory as well as applications in random graph models with local constraints.

Conclusion

This work provides the exact answer for the maximal number of $ex(n,C_6,C_8)$7 copies in $ex(n,C_6,C_8)$8-free graphs for large $ex(n,C_6,C_8)$9, identifying both the count and the unique extremal structure. The methods, anchored in codegree analysis and stability theory, underscore the depth and nuance of extremal combinatorics in the presence of sparse constraints, guiding future investigations into the enumeration of forbidden subgraph configurations.