Tietze's graph

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Tietze's subdivision of a Mobius strip into six mutually-adjacent regions. The vertices and edges of the subdivision form an embedding of Tietze's graph onto the strip. Tietze-Moebius.svg
Tietze's subdivision of a Möbius strip into six mutually-adjacent regions. The vertices and edges of the subdivision form an embedding of Tietze's graph onto the strip.
Tietze's graph
Tietze's graph.svg
The Tietze graph
Vertices 12
Edges 18
Radius 3
Diameter 3
Girth 3
Automorphisms 12 (D6)
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.

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Graph theory study of graphs, which are mathematical structures used to model pairwise relations between objects

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

Cubic graph node-link graphs in which every vertex is incident to exactly three edges

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.

Contents

Relation to Petersen 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.

Petersen graph graph

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.

Triangle shape with three sides

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .

Projective plane Geometric concept of a 2D space with a "point at infinity" adjoined

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect in one and only one point.

Hamiltonicity

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]

In graph theory, a connected graph G is said to be k-vertex-connected if it has more than k vertices and remains connected whenever fewer than k vertices are removed.

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.

Hypohamiltonian graph graph G is said to be hypohamiltonian if G does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from G is Hamiltonian

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

Edge coloring and perfect matchings

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.

Edge coloring an assignment of colors to the edges of a graph so that no two edges that share an endpoint have the same color as each other

In graph theory, an 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 discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.

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. [5]

Snark (graph theory) connected, bridgeless cubic graph with chromatic index equal to 4

In the mathematical field of graph theory, a snark is a simple, connected, bridgeless cubic graph with chromatic index equal to 4. In other words, it is a graph in which every vertex has three neighbors, the connectivity is redundant so that removing no one edge would split the graph, and the edges cannot be colored by only three colors without two edges of the same color meeting at a point. In order to avoid trivial cases, snarks are often restricted to have girth at least 5.

Flower snark

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

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]

Additional properties

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.

See also

Notes

  1. Tietze, Heinrich (1910), "Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen" [Some remarks on the problem of map coloring on one-sided surfaces], DMV Annual Report, 19: 155–159[ permanent dead link ].
  2. 1 2 Clark, L.; Entringer, R. (1983), "Smallest maximally nonhamiltonian graphs", Periodica Mathematica Hungarica, 14 (1): 57–68, doi:10.1007/BF02023582
  3. Weisstein, Eric W. "Tietze's Graph". MathWorld .
  4. Punnim, Narong; Saenpholphat, Varaporn; Thaithae, Sermsri (2007), "Almost Hamiltonian cubic graphs" (PDF), International Journal of Computer Science and Network Security, 7 (1): 83–86
  5. 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 .
  6. 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 Lock-green.svg, doi:10.1002/jgt.21778, MR   3246172 .

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