| Flower snark | |
|---|---|
The flower snarks J3, J5 and J7. |
|
| Vertices | 4n |
| Edges | 6n |
| Girth | 3 for n=3 5 for n=5 6 for n≥7 |
| Chromatic number | 3 |
| Chromatic index | 4 |
| Properties | Snark for n≥5 |
| Notation | Jn with n odd |
| Flower snark J5 | |
|---|---|
The flower snark J5. |
|
| Vertices | 20 |
| Edges | 30 |
| Girth | 5 |
| Chromatic number | 3 |
| Chromatic index | 4 |
| Properties | Snark Hypohamiltonian |
In the mathematical field of graph theory, the flower snarks form an infinite family of snarks introduced by Rufus Isaacs in 1975.[1]
As snarks, the flower snarks are a connected, bridgeless cubic graphs with chromatic index equal to 4. The flower snarks are non-planar and non-hamiltonian.
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The flower snark Jn can be constructed with the following process :
By construction, the Flower snark Jn is a cubic graph with 4n vertices and 6n edges. For it to have the required properties, n should be odd.
The name flower snark is sometimes used for J5, a flower snark with 20 vertices and 30 edges.[2] It is one of 6 snarks on 20 vertices (sequence A130315 in OEIS). The flower snark J5 is hypohamiltonian.[3]
J3 is a trivial variation of the Petersen graph formed by applying a Y-Δ transform to the Petersen graph and thereby replacing one of its vertices by a triangle. This graph is also known as the Tietze's graph.[4] In order to avoid trivial cases, snarks are generally restricted to have girth at least 5. With that restriction, J3 is not a snark.