The Tietze graph
In the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges, formed by applying a Y-Δ transform to the Petersen graph and thereby replacing one of its vertices by a triangle. Tietze's graph has chromatic number 3, chromatic index 4, girth 3 and diameter 3. The independence number is 5. Its automorphism group has order 12, and is isomorphic to the dihedral group D6, the group of symmetries of an hexagon, including both rotations and reflections. This group has two orbits of size 3 and one of size 6 on vertices, and thus this graph is not vertex-transitive.
Like the Petersen graph it is maximally nonhamiltonian: it has no Hamiltonian cycle, but any two non-adjacent vertices can be connected by a Hamiltonian path. It and the Petersen graph are the only 2-vertex-connected cubic non-Hamiltonian graphs with 12 or fewer vertices.
Tietze's graph matches part of the definition of a snark: it is a cubic bridgeless graph that is not 3-edge-colorable. However, some authors restrict snarks to graphs without 3-cycles and 4-cycles, and under this more restrictive definition Tietze's graph is not a snark. Tietze's graph is isomorphic to the graph J3, part of an infinite family of flower snarks introduced by R. Isaacs in 1975.
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