In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles, its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free.
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.
In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k−1 independent sets.
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 are known to 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 Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive.
In the mathematical field of graph theory, the Szekeres snark is a snark with 50 vertices and 75 edges. It was the fifth known snark, discovered by George Szekeres in 1973.
In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter.
In the mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges.
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 the mathematical field of graph theory, the ladder graphLn is a planar undirected graph with 2n vertices and 3n-2 edges.
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. The boundary segments of the regions of Tietze's subdivision form an embedding of Tietze's graph.
In the mathematical field of graph theory, the Frucht graph is a 3-regular graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939.
In the mathematical field of graph theory, the Blanuša snarks are two 3-regular graphs with 18 vertices and 27 edges. They were discovered by Yugoslavian mathematician Danilo Blanuša in 1946 and are named after him. When discovered, only one snark was known—the Petersen graph.
In the mathematical field of graph theory, the Watkins snark is a snark with 50 vertices and 75 edges. It was discovered by John J. Watkins in 1989.
In the mathematical field of graph theory, the double-star snark is a snark with 30 vertices and 45 edges.
In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges.
In the mathematical field of graph theory, the Tutte 12-cage or Benson graph is a 3-regular graph with 126 vertices and 189 edges named after W. T. Tutte.
In the mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte. It has chromatic number 3, chromatic index 3, girth 4 and diameter 8.
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected planar graphs.
