Jones' conjecture

Status partial high confidence

Jones' conjecture — that $cc(G) \leq 2\,cp(G)$ for every planar graph $G$ — remains open for general planar graphs, but has seen substantial partial progress since 2007. The bound for all planar graphs was tightened from $5\,cp(G)$ to $3\,cp(G)$ by Chappell, Gimbel, and Hartman (2014), the conjecture was proved for subcubic planar graphs by Bonamy et al. (2021), and further confirmed for Halin graphs and 'based' planar graphs (those containing a face adjacent to every other face) by Bärnkopf and Győri (2024).

Cited literature (4)

• partial On cycle packings and feedback vertex sets (2014)
  Glenn G. Chappell, John Gimbel, Chris Hartman · Contributions to Discrete Mathematics · doi:10.55016/ojs/cdm.v9i2.62105

  Proves $cc(G) \leq 3\,cp(G)$ for every planar graph (improving the previously known factor-5 bound from Kloks–Lee–Liu), and extends the factor-3 bound to all graphs embedded in surfaces of non-negative Euler characteristic.

• partial Jones' Conjecture in subcubic graphs (2021)
  Marthe Bonamy, François Dross, Tomáš Masařík, Andrea Munaro, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk · The Electronic Journal of Combinatorics · arXiv:1912.01570 · doi:10.37236/9192

  Proves Jones' conjecture in full for subcubic planar graphs: if such a graph has no $k+1$ vertex-disjoint cycles, then $2k$ vertices suffice to destroy all cycles.

• partial Jones' conjecture for Halin graphs and a bit more (2024)
  Pál Bärnkopf, Ervin Győri · arXiv preprint · arXiv:2401.07376

  Proves Jones' conjecture for Halin graphs and, more generally, for 'based planar graphs' — planar graphs that contain a face adjacent to every other face.

• partial Jones' Conjecture in subcubic graphs (2019)
  Marthe Bonamy, François Dross, Tomáš Masařík, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk · arXiv preprint · arXiv:1912.01570

  Proves Jones' Conjecture for all subcubic planar multigraphs, i.e., establishes $\mathrm{fvs}(G) \leq 2 \cdot \mathrm{cp}(G)$ when the maximum degree is at most 3.

Reviewer notes. The Princeton DataSpace entry (dsp01tb09j840d) returned HTTP 401 and could not be verified; it appears to be a thesis on the subcubic case and was not cited. Schlomberg–Thiele–Vygen (arXiv:2207.00450) on packing cycles in planar/bounded-genus graphs was fetched but does not directly address Jones' conjecture and was excluded. The arXiv version of the Bonamy et al. paper (1912.01570, 2019) lists six authors; the published EJC version (2021) adds Andrea Munaro as a seventh — the published list is used here.

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