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Flower snark

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, though they are 1-planar. The flower snarks J<sub>5</sub> and J<sub>7</sub> have book thickness 3 and queue number 2.

Construction

The flower snark J<sub>n</sub> can be constructed with the following process :

  • Build n copies of the star graph on 4 vertices. Denote the central vertex of each star A<sub>i</sub> and the outer vertices B<sub>i</sub>, C<sub>i</sub> and D<sub>i</sub>. This results in a disconnected graph on 4n vertices with 3n edges (A<sub>i</sub> − B<sub>i</sub>, A<sub>i</sub> − C<sub>i</sub> and A<sub>i</sub> − D<sub>i</sub> for 1 ≤ i ≤ n).
  • Construct the n-cycle (B<sub>1</sub>... B<sub>n</sub>). This adds n edges.
  • Finally construct the 2n-cycle (C<sub>1</sub>... C<sub>n</sub>D<sub>1</sub>... D<sub>n</sub>). This adds 2n edges.

By construction, the Flower snark J<sub>n</sub> 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 J<sub>5</sub>, a flower snark with 20 vertices and 30 edges. It is one of 6 snarks on 20 vertices . The flower snark J<sub>5</sub> is hypohamiltonian.

J<sub>3</sub> 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, J<sub>3</sub> is not a snark.

Gallery

References