Flower snark

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Flower snark
Flower snarks.svg
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
Flower snarkv.svg
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 a connected, bridgeless cubic graphs with chromatic index equal to 4. The flower snarks are non-planar and non-hamiltonian.

Construction[edit]

The flower snark Jn can be constructed with the following process :

  • Build n copies of the star graph on 4 vertices. Denote the central vertex of each star Ai and the outer vertices Bi, Ci and Di. This results in a disconnected graph on 4n vertices with 3n edges (Ai-Bi, Ai-Ci and Ai-Di for 1≤in).
  • Construct the n-cycle (B1... Bn). This adds n edges.
  • Finally construct the 2n-cycle (C1... CnD1... Dn). This adds 2n edges.

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[edit]

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

J3 is a trivial variation of the Petersen graph formed by applying a Y-Δ transform to the Petersen graph and thereby 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.

Gallery[edit]

References[edit]

  1. ^ Isaacs, R. "Infinite Families of Nontrivial Trivalent Graphs Which Are Not Tait Colorable." Amer. Math. Monthly 82, 221–239, 1975.
  2. ^ Weisstein, Eric W., "Flower Snark", MathWorld.
  3. ^ Weisstein, Eric W., "Hypohamiltonian Graph", MathWorld.
  4. ^ Clark, L.; Entringer, R. (1983), "Smallest maximally nonhamiltonian graphs", Periodica Mathematica Hungarica 14 (1): 57–68, doi:10.1007/BF02023582 .