Flower snark

In the mathematical field of graph theory, the flower snarks form an infinite family of snarks introduced by Rufus [Isaacs (game theorist)|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 J₅ and J₇ have book thickness 3 and queue number 2.

Construction

The flower snark J_n 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_i and the outer vertices B_i, C_i and D_i. This results in a disconnected graph on 4n vertices with 3n edges.
Construct the n-cycle. This adds n edges.
Finally construct the 2n-cycle. This adds 2n edges.

By construction, the Flower snark J_n 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. It is one of 6 snarks on 20 vertices. 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.