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, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.
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 graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic numberA(G) of a graph G is the fewest colors needed in any acyclic coloring of G.
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than the maximum degree Δ of the graph. At least Δ colors are always necessary, so the undirected graphs may be partitioned into two classes: "class one" graphs for which Δ colors suffice, and "class two" graphs for which Δ + 1 colors are necessary. A more general version of Vizing's theorem states that every undirected multigraph without loops can be colored with at most Δ+µ colors, where µ is the multiplicity of the multigraph. The theorem is named for Vadim G. Vizing who published it in 1964.
In the mathematical field of graph theory, the Pappus graph is a bipartite 3-regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the cubic distance-regular graphs are known; the Pappus graph is one of the 13 such graphs.
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch, who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable.
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 theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that
In the mathematical field of graph theory, the Chvátal graph is an undirected graph with 12 vertices and 24 edges, discovered by Václav Chvátal in 1970. It is the smallest graph that is triangle-free, 4-regular, and 4-chromatic.
In the mathematical field of graph theory, the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges.
In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3-regular, undirected graph with 70 vertices and 105 edges.
In the mathematical field of graph theory, the Harries–Wong graph is a 3-regular undirected graph with 70 vertices and 105 edges.
In the mathematical field of graph theory, the McGee graph or the (3-7)-cage is a 3-regular graph with 24 vertices and 36 edges.
In the mathematical field of graph theory, the Franklin graph is a 3-regular graph with 12 vertices and 18 edges.
In the mathematical field of graph theory, the Folkman graph is a 4-regular graph with 20 vertices and 40 edges. It is a regular bipartite graph with symmetries taking every edge to every other edge, but the two sides of its bipartition are not symmetric with each other, making it the smallest possible semi-symmetric graph. It is named after Jon Folkman, who constructed it for this property in 1967.
In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson.
In the mathematical field of graph theory, the Horton graph or Horton 96-graph is a 3-regular graph with 96 vertices and 144 edges discovered by Joseph Horton. Published by Bondy and Murty in 1976, it provides a counterexample to the Tutte conjecture that every cubic 3-connected bipartite graph is Hamiltonian.
In the mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges. It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph. The same graph had already been given as an example of a non-Hamiltonian simplicial polyhedron by Branko Grünbaum in 1967.
