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
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
The flower snark J5.
Vertices	20
Edges	30
Girth	5
Chromatic number	3
Chromatic index	4
Properties	Snark Hypohamiltonian
Table of graphs and parameters

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, though they are 1-planar. ^[2] The flower snarks J₅ and J₇ have book thickness 3 and queue number 2. ^[3]

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 (A_i − B_i, A_i − C_i and A_i − D_i for 1 ≤ i ≤ n).
• Construct the n-cycle (B₁... B_n). This adds n edges.
• Finally construct the 2n-cycle (C₁... C_nD₁... D_n). 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. ^[4] It is one of 6 snarks on 20 vertices (sequence A130315 in the OEIS ). The flower snark J₅ is hypohamiltonian. ^[5]

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. ^[6] In order to avoid trivial cases, snarks are generally restricted to have girth at least 5. With that restriction, J₃ is not a snark.

Gallery

• 
  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.
• 
  The Petersen graph as a graph minor of the flower snark J₅

References

1. Isaacs, R. (1975). "Infinite Families of Nontrivial Trivalent Graphs Which Are Not Tait Colorable". Amer. Math. Monthly. 82 (3): 221–239. doi:10.1080/00029890.1975.11993805. JSTOR   2319844.
2. Pupyrev, Sergey (2025), "OOPS: Optimized One-Planarity Solver via SAT", in Dujmović, Vida; Montecchiani, Fabrizio (eds.), Proc. 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025), Leibniz International Proceedings in Informatics (LIPIcs), vol. 357, pp. 14:1–14:19, doi: 10.4230/LIPIcs.GD.2025.14 , ISBN   978-3-95977-403-1 .
3. Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
4. Weisstein, Eric W. "Flower Snark". MathWorld .
5. Weisstein, Eric W. "Hypohamiltonian Graph". MathWorld .
6. Clark, L.; Entringer, R. (1983), "Smallest maximally nonhamiltonian graphs", Periodica Mathematica Hungarica, 14 (1): 57–68, doi:10.1007/BF02023582 .