Flower snark
| 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.
As snarks, the flower snarks are connected, bridgeless cubic graphs with chromatic index equal to 4. The flower snarks are non-planar and non-hamiltonian.
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. It is one of 6 snarks on 20 vertices (sequence in the OEIS). The flower snark J5 is hypohamiltonian.
J3 is a trivial variation of the Petersen graph formed by replacing one of its vertices by a triangle. This graph is also known as the Tietze's graph. In order to avoid trivial cases, snarks are generally restricted to have girth at least 5. With that restriction, J3 is not a snark.
The chromatic number of the flower snark J5 is 3.
The chromatic index of the flower snark J5 is 4.
The original representation of the flower snark J5.
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Wikipedia