Let G) be a 2-connected non-bipartite graph with minimum degree at least k+1. Liu–Ma conjecture. G contains eil k/2 cycles with consecutive odd lengths. Liu and Ma proposed this conjecture after proving the lower bound floor k/2; the conjecture is sharp for the complete graph K_k+2. This paper's abstract states that the conjecture is confirmed for every k∈ N, so the conjecture is solved.

Liu–Ma conjecture on consecutive odd cycles

Liu and Ma conjectured that every qualifying graph contains the maximum possible number of cycles whose odd lengths are consecutive. The conjecture is sharp for complete graphs.

Known results

Liu and Ma proved a lower bound of $\lfloor k/2\rfloor$ consecutive odd cycles.
The complete graph $K_{k+2}$ shows sharpness at $\lceil k/2\rceil$ .

October 1, 2024 claimed proof

A preprint, A strengthening on consecutive odd cycles in graphs of given minimum degree, states that Liu–Ma’s conjecture is confirmed for every $k\in\mathbb{N}$ . Its Theorem 1.1 asserts the conjectured $\lceil k/2\rceil$ cycles for every $k\geq 1$ , but the scan found no independent verification or referee report.

Current status (as of September 2026): The conjecture has a published preprint claiming a complete proof for every $k\geq 1$ , but that claim remains unverified; no counterexample or documented error was found.

Liu–Ma conjecture on consecutive odd cycles

Primary source

Progress summary

Solutions 0