# Jones' conjecture

For a graph , let denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let denote the cardinality of a minimum feedback vertex set (set of vertices so that is acyclic).

**Conjecture**For every planar graph , .

In [KLL], the authors mention that there exists a family of nonplanar graphs for which , so no such result could hold for general graphs. They also point out that the conjecture is tight for wheels, and they prove it for the special case of outerplanar graphs.

## Bibliography

*[KLL] Ton Kloks, Chuan-Min Lee, and Jiping Liu, New Algorithms for -Face Cover, -Feedback Vertex Set, and -Disjoint Cycles on Plane and Planar Graphs, in *Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science* (WG2002), LNCS 2573, pp. 282--295, 2002.

* indicates original appearance(s) of problem.

### Why Jones'?

Does anyone know why this is called Jones' Conjecture?

### Reply: Why Jones'?

I am Jones. My Taiwanese name is Chuan-Min Lee. This conjecture came up when I was working on it with Ton Kloks and Jiping Liu. I used the name "Jones" instead of my Taiwanese name for ease of communication.

## Proved for subcubic planar

Proved for subcubic planar graphs by Marthe Bonamy, François Dross, Tomáš Masařík, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk [https://arxiv.org/abs/1912.01570].