In the mathematical field of graph theory, the flower snarks form an infinite family of snarks introduced by Rufus Isaacs in 1975.[1]
Flower snark | |
---|---|
Vertices | 4n |
Edges | 6n |
Girth | 3 for n=3 5 for n=5 6 for n≥7 |
Chromatic number | 3 |
Chromatic index | 4 |
Book thickness | 3 for n=5 3 for n=7 |
Queue number | 2 for n=5 2 for n=7 |
Properties | Snark for n≥5 |
Notation | Jn with n odd |
Table of graphs and parameters |
Flower snark J5 | |
---|---|
Vertices | 20 |
Edges | 30 |
Girth | 5 |
Chromatic number | 3 |
Chromatic index | 4 |
Properties | Snark Hypohamiltonian |
Table of graphs and parameters |
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 snarks J5 and J7 have book thickness 3 and queue number 2.[2]
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.[3] It is one of 6 snarks on 20 vertices (sequence A130315 in the OEIS). The flower snark J5 is hypohamiltonian.[4]
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.[5] In order to avoid trivial cases, snarks are generally restricted to have girth at least 5. With that restriction, J3 is not a snark.