Tietze's graph | |
---|---|

Vertices | 12 |

Edges | 18 |

Radius | 3 |

Diameter | 3 |

Girth | 3 |

Automorphisms | 12 (D_{6}) |

Chromatic number | 3 |

Chromatic index | 4 |

Properties | Cubic Snark |

Table of graphs and parameters |

In the mathematical field of graph theory, **Tietze's graph** is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz Friedrich Tietze, who showed in 1910 that the Möbius strip can be subdivided into six regions that all touch each other – three along the boundary of the strip and three along its center line – and therefore that graphs that are embedded onto the Möbius strip may require six colors.^{ [1] } The boundary segments of the regions of Tietze's subdivision (including the segments along the boundary of the Möbius strip itself) form an embedding of Tietze's graph.

Tietze's graph may be formed from the Petersen graph by replacing one of its vertices with a triangle.^{ [2] }^{ [3] } Like the Tietze graph, the Petersen graph forms the boundary of six mutually touching regions, but on the projective plane rather than on the Möbius strip. If one cuts a hole from this subdivision of the projective plane, surrounding a single vertex, the surrounded vertex is replaced by a triangle of region boundaries around the hole, giving the previously described construction of the Tietze graph.

Both Tietze's graph and the Petersen graph are *maximally nonhamiltonian*: they have no Hamiltonian cycle, but any two non-adjacent vertices can be connected by a Hamiltonian path.^{ [2] } Tietze's graph and the Petersen graph are the only 2-vertex-connected cubic non-Hamiltonian graphs with 12 or fewer vertices.^{ [4] }

Unlike the Petersen graph, Tietze's graph is not hypohamiltonian: removing one of its three triangle vertices forms a smaller graph that remains non-Hamiltonian.

Edge coloring Tietze's graph requires four colors; that is, its chromatic index is 4. Equivalently, the edges of Tietze's graph can be partitioned into four matchings, but no fewer.

Tietze's graph matches part of the definition of a snark: it is a cubic bridgeless graph that is not 3-edge-colorable. However, most authors restrict snarks to graphs without 3-cycles, so Tietze's graph is not generally considered to be a snark. Nevertheless, it is isomorphic to the graph J_{3}, part of an infinite family of flower snarks introduced by R. Isaacs in 1975.^{ [5] }

Unlike the Petersen graph, the Tietze graph can be covered by four perfect matchings. This property plays a key role in a proof that testing whether a graph can be covered by four perfect matchings is NP-complete.^{ [6] }

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 D_{6}, the group of symmetries of a 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.

- The chromatic number of the Tietze graph is 3.
- The chromatic index of the Tietze graph is 4.
- The Tietze graph has crossing number 2 and is 1-planar.
- A three-dimensional embedding of the Tietze graph.

Wikimedia Commons has media related to Tietze's graph .

- Dürer graph and Franklin graph, two other 12-vertex cubic graphs

- ↑ Tietze, Heinrich (1910), "Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen" [Some remarks on the problem of map coloring on one-sided surfaces](PDF),
*DMV Annual Report*,**19**: 155–159 - 1 2 Clark, L.; Entringer, R. (1983), "Smallest maximally nonhamiltonian graphs",
*Periodica Mathematica Hungarica*,**14**(1): 57–68, doi:10.1007/BF02023582 - ↑ Weisstein, Eric W. "Tietze's Graph".
*MathWorld*. - ↑ Punnim, Narong; Saenpholphat, Varaporn; Thaithae, Sermsri (2007), "Almost Hamiltonian cubic graphs" (PDF),
*International Journal of Computer Science and Network Security*,**7**(1): 83–86 - ↑ Isaacs, R. (1975), "Infinite families of nontrivial trivalent graphs which are not Tait colorable",
*Amer. Math. Monthly*, Mathematical Association of America,**82**(3): 221–239, doi:10.2307/2319844, JSTOR 2319844 . - ↑ Esperet, L.; Mazzuoccolo, G. (2014), "On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings",
*Journal of Graph Theory*,**77**(2): 144–157, arXiv: 1301.6926 , doi:10.1002/jgt.21778, MR 3246172 .

In the mathematical field of graph theory, the **Petersen graph** is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring.

This is a **glossary of graph theory**. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

In graph theory, an **outerplanar graph** is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.

In graph theory, a **proper edge coloring** of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The **edge-coloring problem** asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the **chromatic index** of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

In the mathematical field of graph theory, a **snark** is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three colors. In order to avoid trivial cases, snarks are often restricted to have additional requirements on their connectivity and on the length of their cycles. Infinitely many snarks exist.

In the mathematical field of graph theory, a **cubic graph** is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called **trivalent graphs**.

In the mathematical field of graph theory, the **Heawood graph** is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.

In the mathematical field of graph theory, the **Desargues graph** is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases.

In the mathematical field of graph theory, the **Wagner graph** is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph.

In the mathematical field of graph theory, a graph *G* is said to be **hypohamiltonian** if *G* itself does not have a Hamiltonian cycle but every graph formed by removing a single vertex from *G* is Hamiltonian.

In graph-theoretic mathematics, a **cycle double cover** is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide a double cover of the graph: each edge belongs to exactly two faces.

In the mathematical field of graph theory, the **Dürer graph** is an undirected graph with 12 vertices and 18 edges. It is named after Albrecht Dürer, whose 1514 engraving *Melencolia I* includes a depiction of Dürer's solid, a convex polyhedron having the Dürer graph as its skeleton. Dürer's solid is one of only four well-covered simple convex polyhedra.

In graph theory, the **generalized Petersen graphs** are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins.

In the mathematical field of graph theory, the **flower snarks** form an infinite family of snarks introduced by Rufus Isaacs in 1975.

In the mathematical field of graph theory, the **Nauru graph** is a symmetric, bipartite, cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the twelve-pointed star in the flag of Nauru.

In the mathematical field of graph theory, the **double-star snark** is a snark with 30 vertices and 45 edges.

In combinatorial mathematics, an **Apollonian network** is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.

In the mathematical discipline of graph theory, **Petersen's theorem**, named after Julius Petersen, is one of the earliest results in graph theory and can be stated as follows:

Petersen's Theorem.Every cubic, bridgeless graph contains a perfect matching.

* The Petersen Graph* is a mathematics book about the Petersen graph and its applications in graph theory. It was written by Derek Holton and John Sheehan, and published in 1993 by the Cambridge University Press as volume 7 in their Australian Mathematical Society Lecture Series.

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