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.[1]

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.

Construction

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.

Special cases

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 the OEIS). The flower snark J5 is hypohamiltonian.[3]

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.[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.

References

  1. Isaacs, R. "Infinite Families of Nontrivial Trivalent Graphs Which Are Not Tait Colorable." Amer. Math. Monthly 82, 221239, 1975.
  2. Weisstein, Eric Wolfgang. "Flower Snark". MathWorld.
  3. Weisstein, Eric Wolfgang. "Hypohamiltonian Graph". MathWorld.
  4. Clark, L.; Entringer, R. (1983), "Smallest maximally nonhamiltonian graphs", Periodica Mathematica Hungarica, 14 (1): 57–68, doi:10.1007/BF02023582.
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