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OP-9302OpenGraph theory

Erdős #1016 · graph theory / cycles

Let h(n)h(n) be minimal such that there is a graph on nn vertices with n+h(n)n+h(n) edges which contains a cycle on kk vertices, for all 3≤k≤n3\leq k\leq n. Estimate h(n)h(n). In particular, is it true that

h(n)≥log⁡2n+…h(n) \geq \log_2n+…
MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let h(n)h(n) be minimal such that there is a graph on nn vertices with n+h(n)n+h(n) edges which contains a cycle on kk vertices, for all 3≤k≤n3\leq k\leq n. Estimate h(n)h(n). In particular, is it true that

h(n)≥log⁡2n+log⁡∗n−O(1),h(n) \geq \log_2n+\log_*n-O(1),

where log⁡∗n\log_*n is the iterated logarithmic function?

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Background

Such graphs are called pancyclic. A problem of Bondy [Bo71], who claimed a proof (without details) of

log⁡2(n−1)−1≤h(n)≤log⁡2n+log⁡∗n+O(1).\log_2(n-1)-1\leq h(n) \leq \log_2n+\log_*n+O(1).

Erdős [Er71] believed the upper bound is closer to the truth, but could not even prove h(n)−log⁡2n→∞h(n)-\log_2n\to \infty.

A proof of the above lower bound is provided by Griffin [Gr13]. The first published proof of the upper bound appears to be in Chapter 4.5 of George, Khodkar, and Wallis [GKW16].

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1016 link

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