1 Proof of Theorem 1
A multigraph is simple if it has no loops or multi-edges. In this case, we simply refer to it as a graph.
Let be a (multi)graph. For , we denote by the subgraph of induced by , and by the subgraph of induced by . If , then we denote . For , we denote . If , then we denote . For , is the disjoint union of and a set of isolated vertices. If , then we denote . For a set of pairs of edges with , we denote . If , then we denote . Graph is called cubic if all of its degrees are exactly . Graph is called subcubic if its degrees are not bigger than .
A (multi)graph is -connected if the removal of any vertices leaves the graph connected. A (multi)graph is -edge-connected if the removal of at most edges leaves the graph connected. Note that a subcubic (multi)graph with at least vertices is -connected if and only if it is -edge-connected.
A (multi)graph is essentially -edge-connected if the removal of at most three edges does not yield two components with at least two vertices each. A (multi)graph is cyclically -edge-connected if the removal of at most three edges does not yield two components that both contain a cycle. For a cubic (multi)graph, these last two notions are equivalent.
Theorem 2 (Theorem 3.4.10 in [Mun16]).
If is simple, then it is not cyclically 4-edge-connected.
To obtain a contradiction, we argue that is in fact, a simple graph that is essentially -edge-connected, as follows.
The multigraph is an essentially -edge-connected simple graph.
While Claims 1, 2, 3 are known and easy properties of a minimum counter-example to Jones’ conjecture on subcubic graphs (see e.g. [Mun16]), we include their proofs because we believe they may constitute a useful warm-up. The uninterested reader may skip them guilt-free.
The multigraph is -regular.
Suppose has a vertex with degree at most . Then satisfies Jones’ Conjecture by minimality of . As no cycle of contains , we have and , therefore also satisfies Jones’ Conjecture, a contradiction.
Suppose has a vertex with degree , and let and be the two neighbors of . Then satisfies Jones’ Conjecture, so . The cycles of are in bijection with the cycles of , by exchanging the edges and and an edge when appropriate. Hence . Moreover, if is a feedback vertex set of that does not contain , then is a feedback vertex set of , and if is a feedback vertex set of that contains , then is a feedback vertex set of . Thus , and satisfies Jones’ Conjecture, a contradiction. Hence is cubic.
The multigraph is -connected.
Suppose that is not -connected. As is cubic, that means that is not -edge-connected. Let be a separating edge of . Both components and of verify Jones’ Conjecture by minimality of . Since is separating, it is not in any cycle of . The union of any feedback vertex set of and any feedback vertex set of is a feedback vertex set of , so . The union of any cycle packing of and any cycle packing of is a cycle packing of , so . Therefore satisfies Jones’ Conjecture, a contradiction.
In particular since is cubic, Claim 2 implies that is a simple graph.
The graph is -connected.
Assume that it is not -connected, and thus not -edge-connected. Let and be a -edge-cut, where and are in the same connected component of , which we denote . Let be the other connected component of . We write and . Note that this may lead to a double edge. See Figure 1 for an illustration. By minimality of , we know that , , and all satisfy Jones’ Conjecture.
Note that since , we have . We first argue that . Assume for a contradiction that . Note that for any feedback vertex set of , either or or and are in distinct components of , so . Thus, , a contradiction.
By symmetry, we have . Therefore, every cycle packing of contains the edge and every cycle packing of contains the edge . We can thus combine a cycle packing of and a cycle packing of by making a single cycle out of those two cycles. So . However, if is a feedback vertex set of and is a feedback vertex set of , then is a feedback vertex set of . Therefore , a contradiction. Therefore is -connected.
The graph is essentially -edge-connected.
Assume that is not essentially -edge-connected, and thus not cyclically -edge-connected. Consider a non-trivial -edge-cut . Let and be the two components of . For , we define as the graph obtained from by contracting into a single vertex . See Figure 2 for an illustration. We define (resp. , ) as the graph obtained from by connecting with an edge vertices from incident to and (resp. to and for or to and for ). Again, this may lead to a double edge. Note that for both values of , all of , , , and have fewer vertices than , and thus satisfy Jones’ Conjecture.
First note that for both values of , . In order to prove that let us assume without loss of generality that . Then remove from . What remains from after deleting is a forest that could hypothetically create connections between vertices from incident to . However if we are given any tree and its three vertices then (in fact it is always a single vertex), where are sets of vertices on unique paths between corresponding vertices, so it is possible to break the connections between all three pairs of these vertices by removing a single vertex of . Because of that we see that what leads to . Therefore we see that . We also have , yet .
It follows that for both values of :
We are now ready for a closer analysis.
For any and for any , we have
Indeed, take without loss of generality and . Let us consider a minimum feedback vertex set of . Note that in , there is no path between the vertex incident to and the vertex incident to or at least one of them is in . As a consequence, either vertex incident to is in or one of them, say the vertex incident to , is either in or is not in the same component as the vertex incident to . For any minimum feedback vertex set of , we observe that is a feedback vertex set of , hence the conclusion. In particular, by combining with (3), if then for some .
For every and for every , if , then . That follows from (2), since satisfies Jones’ Conjecture.
For every , we have either or . Indeed, suppose not. Then both and , for say . Then for , in every cycle packing of there is a cycle containing and . By taking a cycle packing of and a cycle packing of , we obtain a cycle packing of (combining two cycles into one). So , a contradiction with (4).
We assume without loss of generality that . Note that from (iii), , hence , and by (ii). We assume without loss of generality . By symmetry, and . From (i) applied with and , there is such that . Note that , so and . We derive from (ii) that , hence by (iii). Again from (ii), we obtain .
Therefore we have: , , and . Now, (i) applied with and yields a contradiction.
Through a non-trivial combination of elementary tricks and using a nice preliminary result of [Mun16], we were able to close the case of Jones’ Conjecture for subcubic graphs.
The obvious question is whether this can be at all used to solve the whole conjecture. The reduction we have for subcubic graphs extends easily to the general setting, in the sense that a smallest counter-example to Jones’ Conjecture is essentially -edge-connected. It is not difficult to argue in a similar way that such a graph is -vertex-connected. However, a much harder question is whether it is essentially -vertex-connected. While it still seems possible, such a result using our approach would require additional tricks. Note that being in the general setting also gives us more leeway regarding possible reductions (no need to shy away from increasing the maximum degree, as long as there are fewer vertices).
A second obstacle to generalization is that even assuming that a smallest counter-example is essentially -vertex-connected, Theorem 2 only deals with the subcubic case. Another argument must then be devised.
A different approach would be not to aim for the conjectured bound of but simply for any bound better than the existing one of . Unfortunately, this does not seem conceptually much easier. Let us emphasize this: a simple discharging argument yields for every planar graph , while even significant effort fails to grant a factor of instead of .
To highlight how little we understand around Jones’ Conjecture, we conclude by posing the following stronger conjecture. Note that the example of many nested disjoint cycles shows that the embedding cannot be fixed. Also note that the simple discharging argument mentioned above does not imply the following conjecture with a factor of instead of .
For every planar graph , we have
where is the maximum size of a face-packing of , i.e., a cycle-packing where, for some embedding of , every cycle bounds a face.
- [CFS12] Hong-Bin Chen, Hung-Lin Fu, and Chie-Huai Shih. Feedback vertex set on planar graphs. Taiwanese Journal of Mathematics, 16(6):2077–2082, 2012.
- [CGH14] Glenn G. Chappell, John Gimbel, and Chris Hartman. On cycle packings and feedback vertex sets. Contributions to Discrete Mathematics, 9(2), 2014.
- [CvBHJR92] Wouter Cames van Batenburg, Tony Huynh, Gwenaël Joret, and Jean-Florent Raymond. A tight Erdős-Pósa function for planar minors. Advances in Combinatorics, 33pp, 2019:2.
- [EP65] Paul Erdős and Lajos Pósa. On independent circuits contained in a graph. Canadian Journal of Mathematics, 17:347–352, 1965.
- [KLL02] Ton Kloks, Chuan-Min Lee, and Jiping Liu. New algorithms for k-face cover, k-feedback vertex set, and k-disjoint cycles on plane and planar graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 282–295. Springer, 2002.
- [Mun16] Andrea Munaro. Sur quelques invariants classiques et nouveaux des hypergraphes. PhD thesis, Grenoble Alpes, 2016.
Jie Ma, Xingxing Yu, and Wenan Zang.
Approximate min-max relations on plane graphs.
Journal of Combinatorial Optimization, 26(1):127–134, 2013.
- [RS86] Neil Robertson and Paul D. Seymour. Graph minors. V. excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92–114, 1986.