An improved upper bound for the planar Turán number of $C_8$

Abstract: We prove that every $n$-vertex simple planar graph with no copy of $C_8$ has at most [ \frac{69}{25}(n-2) ] edges, for every $n\ge 8$. This improves the best known bound [ \frac{323}{108}n-6 \qquad \text{for every } n\ge 27. ]

• The paper proves that every n-vertex simple planar graph without a copy of C₈ has at most 69/25(n−2) edges, improving the previous leading coefficient from approximately 2.991 to 2.76.
• The proof combines Euler-based discharging with six exhaustive, reproducible computer-certified analyses of finite plane patches, including a large-face load bound and an eight-vertex base-case verification.
• The result narrows but does not close the gap with the lower bound of 21/8n−27/4, leaving the exact planar Turán number of C₈ and the conjectured leading coefficient of 21/8 unresolved.

Overview and main result

This paper establishes that every $n$-vertex simple planar graph with no copy of $C_8$ (not necessarily induced) has at most $\frac{69}{25}(n-2)$ edges, for all $n \ge 8$. This improves the previous best bound of $\frac{323}{108}n - 6$, valid for $n \ge 27$, which followed from the general theta-graph estimate of Shi, Walsh, and Yu. The improvement is substantial in the leading coefficient: $69/25 = 2.76$ versus $323/108 \approx 2.991$.

The authors are careful to state what the result does not claim: the exact planar Turán number is not determined, equality cases are not characterized, and no claim is made that $69/25$ is best possible. The best known lower bound comes from a construction of Cranston, Lidický, Liu, and Shantanam giving $ex(n,C_8) \ge \frac{21}{8}n - \frac{27}{4}$ for infinitely many $C_8$0, so a gap between coefficients $C_8$1 and $C_8$2 remains.

Context: planar Turán numbers of cycles

The planar Turán number $C_8$3 is the maximum edge count among $C_8$4-vertex simple planar graphs avoiding $C_8$5 as a subgraph. For cycles, the known upper bounds are:

Cycle	Upper bound	Source
$C_8$6	$C_8$7	Euler's formula
$C_8$8	$C_8$9	Dowden
$\frac{69}{25}(n-2)$0	$\frac{69}{25}(n-2)$1	Dowden
$\frac{69}{25}(n-2)$2	$\frac{69}{25}(n-2)$3	Ghosh–Győri–Martin–Paulos–Xiao
$\frac{69}{25}(n-2)$4	$\frac{69}{25}(n-2)$5 (tight)	Shi–Walsh–Yu
$\frac{69}{25}(n-2)$6	$\frac{69}{25}(n-2)$7	this paper

Ghosh et al. conjectured that for each $\frac{69}{25}(n-2)$8 and all sufficiently large $\frac{69}{25}(n-2)$9, $n \ge 8$0; for $n \ge 8$1 this predicts leading coefficient $n \ge 8$2. Cranston et al. disproved the conjecture for all $n \ge 8$3, but their counterexamples do not settle $n \ge 8$4, which remains open. Notably, the paper's new coefficient $n \ge 8$5 exceeds the conjectured $n \ge 8$6, so the conjectured value for $n \ge 8$7 remains consistent with this upper bound but is not confirmed by it.

Method: discharging supported by finite local certificates

The proof combines an inductive discharging argument with six computer-certified local facts about plane patches. After standard reductions (disconnected graphs, cut vertices, vertices of degree at most two are handled by induction), any minimal counterexample is simple, 2-connected, $n \ge 8$8-free, has minimum degree at least three, and has $n \ge 8$9.

Each face receives initial charge $\frac{323}{108}n - 6$0, summing to $\frac{323}{108}n - 6$1 by Euler's formula. With $\frac{323}{108}n - 6$2, each face sends $\frac{323}{108}n - 6$3 to each incident vertex, leaving residual face charge $\frac{323}{108}n - 6$4. Triangular faces then receive compensation from their "nearest" $\frac{323}{108}n - 6$5-faces: each triangle carries one unit of load split equally among its nearest $\frac{323}{108}n - 6$6-faces (those minimizing dual distance), and each such face $\frac{323}{108}n - 6$7 sends $\frac{323}{108}n - 6$8 to each contributing triangle $\frac{323}{108}n - 6$9. Triangles end with charge exactly zero since $n \ge 27$0.

The key structural input (Lemma on covering by nearby $n \ge 27$1-faces) is that every triangular face has a $n \ge 27$2-face within dual distance $n \ge 27$3 — otherwise the radius-three all-triangular neighbourhood would force the whole graph to have at most seven vertices, contradicting $n \ge 27$4. This ensures the load distribution is well defined and bounded.

The local bounds

The discharging closes using four certified load bounds:

• Faces of degree 4: the equal-split load satisfies $n \ge 27$5. This follows from the certified facts that a 4-face has at most seven bad contributors, at most six unique-nearest contributors, and in the dangerous case (six unique-nearest plus one further contributor) the further contributor has at least three nearest faces, contributing at most $n \ge 27$6.
• Faces of degree 5, 6, 7: bad counts are at most $n \ge 27$7 respectively, yielding final charges $n \ge 27$8, $n \ge 27$9, $69/25 = 2.76$0.
• Degree 8: impossible, since the boundary would itself be an 8-cycle.
• Degree $69/25 = 2.76$1: the per-edge load satisfies $69/25 = 2.76$2 for every boundary edge, giving $69/25 = 2.76$3 and final charge $69/25 = 2.76$4.

All faces therefore end with nonnegative charge, and charge conservation yields $69/25 = 2.76$5.

The base case $69/25 = 2.76$6 uses a separate certificate: no planar non-Hamiltonian graph on eight vertices has 17 or 18 edges. Since a Hamiltonian cycle on eight vertices is exactly a $69/25 = 2.76$7, every 8-vertex planar graph exceeding $69/25 = 2.76$8 edges contains a copy of $69/25 = 2.76$9.

The computer-assisted verification

The six certificates (C1)–(C6) are established by exhaustive enumeration of finite rooted plane patches, with full reproducibility infrastructure described in the appendix. The total verification suite runs in about 14.3 minutes. Several design points deserve emphasis:

Lifting and domination. A lifting-and-domination proposition shows that every genuine local configuration arising in a simple 2-connected $323/108 \approx 2.991$0-free plane graph with $323/108 \approx 2.991$1 occurs in the corresponding finite search, and that the searches may retain extra states without harm since only upper bounds are needed. Rejection rules (repeated triangles, triple facial incidences, non-planar boundary order, actual 8-cycles, shared two-edge facial paths under minimum degree three) are all necessary conditions.

The sealed search for large faces. Certificate (C5) is the most involved. The verifier roots at a three-edge window of the root face boundary, compresses the omitted arc into ordered "virtual sides" (bookkeeping objects, not graph edges), and branches over triangular continuations, $323/108 \approx 2.991$2-seals (non-root $323/108 \approx 2.991$3-faces), and $323/108 \approx 2.991$4-seals (other root entries). A faithful cut-open model lemma proves injectivity of vertex labels and that restoring unlisted portions can only enlarge denominators or zero out contributions, so terminal values dominate true loads. The search tree contains 223,766 labelled states and 136,987 terminal leaves; a C++ checker re-parses and re-verifies the archived certificate, including audits confirming zero construction-layer underestimates, zero omitted relevant branches above layer six, and zero invalid boundary components. Every terminal state has computed load at most $323/108 \approx 2.991$5.

Base case enumeration. Certificate (C6) enumerates all labelled 8-vertex graphs with 17 or 18 edges (21,474,180 and 13,123,110 graphs respectively), filters by Hamiltonicity, applies induced-subgraph sparsity, and runs exhaustive memoized deletion–contraction $323/108 \approx 2.991$6 minor tests per Wagner's theorem. Zero planar non-Hamiltonian graphs remain.

The declaration notes that OpenAI Codex was used only to assist in implementing and checking the programs; the mathematical lifting arguments are proved by hand in the text.

Limitations and open questions

Several limitations are stated plainly. First, the result is an upper bound only: the gap between the lower-bound coefficient $323/108 \approx 2.991$7 and the upper-bound coefficient $323/108 \approx 2.991$8 persists, and the exact value of $323/108 \approx 2.991$9 remains undetermined. Second, the proof depends on computer-assisted finite verifications; while certificates, source code, and reproduction scripts are publicly available, the correctness of the result rests on these computations and their audit tooling rather than purely hand-checkable arguments. Third, the method relies on the specific structure of $69/25$0 (in particular, degree-8 faces being forbidden outright); whether the same certificate-based discharging framework extends to longer cycles, where Cranston et al.'s counterexamples show the Ghosh et al. conjecture fails for $69/25$1, is not addressed. Finally, the conjectured tight coefficient $69/25$2 for $69/25$3 remains neither proved nor refuted.

Conclusion

The paper improves the best known upper bound for the planar Turán number of $69/25$4 from approximately $69/25$5 to $69/25$6 via a discharging argument whose local inputs are certified by exhaustive, reproducible finite computations. The verification infrastructure — machine-readable certificates, independent re-checking programs, and explicit lifting lemmas connecting finite patches to arbitrary ambient graphs — is a notable feature of the presentation. The central open question left by the work is whether the true leading coefficient equals the conjectured $69/25$7, or lies strictly between $69/25$8 and $69/25$9.