The sets X and Y are called the domain ( or the set of departure ) and codomain ( or the set of destination ), respectively, of the relation, and G is called its graph.

The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).

Call this graph G. G cannot have a vertex of degree 3 or less, because if d ( v ) ≤ 3, we can remove v from G, four-color the smaller graph, then add back v and extend the four-coloring to it by choosing a color different from its neighbors.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

In graph theory, an isomorphism between two graphs G and H is a bijective map f from the vertices of G to the vertices of H that preserves the " edge structure " in the sense that there is an edge from vertex u to vertex v in G if and only if there is an edge from ƒ ( u ) to ƒ ( v ) in H. See graph isomorphism.

* How many graph colorings using k colors are there for a particular graph G?

Namely, any free group G may be realized as the fundamental group of a graph X.

The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X ; but every such Y is again a graph.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

We must now show that on some component of the graph there exist two points for which the corresponding diagonal points in the C-plane are on opposite sides of C.

It should be noted that while there are very strong connections between graph theory and combinatorics, these two are sometimes thought of as separate subjects.

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

Although there is a downward population curve, explained by a larger death than birth rate, as well as a larger number of emigrants than immigrants, the line graph of the natural population increase shows the rate of population decrease was slowly diminishing.

If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable.

In general, the surrounding graph must be systematically recolored to turn the ring's coloring into a good one, as was done in the case above where there were 4 neighbors ; for a general configuration with a larger ring, this requires more complex techniques.

For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries-Wong graph.

For any positive integers and, there exists a graph with girth at least and chromatic number at least ; for instance, the Grötzsch graph is triangle-free and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number.

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 ( mathematics ) for more detailed definitions and for other variations in the types of graph that are commonly considered.

A graph drawing should not be confused with the graph itself ( the abstract, non-visual structure ) as there are several ways to structure the graph drawing.

( However, there are other, similar matrices that are also called " Laplacian matrices " of a graph.

The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi tessellation for P. Special cases include the existence of three points on a line and four points on circle.

This graph is planar ( it is important to note that we are talking about the graphs that have some limitations according to the map they are transformed from only ): it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves that lead without crossing within each region from the vertex location to each shared boundary point of the region.

To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices.

A preorder that is antisymmetric no longer has cycles ; it is a partial order, and corresponds to a directed acyclic graph.

Integration is a mathematical operation that corresponds to the informal idea of finding the area under the graph of a function.

If y is directly proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.

The sequential unfolding of modules is observed as a very characteristic sawtooth pattern of the force vs elongation graph ; every tooth corresponds to the unfolding of a single protein module ( apart from the last that is generally the detachment of the protein molecule from the tip ) A lot of information about protein elasticity and protein unfolding can be obtained by this technique.

At age 117, she also set the record for the world's oldest " new " title-holder ( which corresponds to the highest " valley " on a graph of the oldest living persons over time ).

The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph.

Such a program corresponds to a finite state machine ( FSM ), i. e., a directed graph consisting of nodes ( or vertices ) and edges.

The partition of the Turán graph into independent sets corresponds to the partition of G into color classes.

The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative ; then the column is all zero except for ± 2 in the row of its incident vertex.

In this way, the graph of a continuous real-valued function in the plane corresponds to an infinite set of basis functions ; if the number of basis functions were finite, the curve would consist of a discrete set of points rather than a continuous contour.

The set of all solutions to a 2-satisfiability instance has the structure of a median graph, in which an edge corresponds to the operation of flipping the values of a set of variables that are all constrained to be equal or unequal to each other.

The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph.

An important example of this type comes from computational geometry: the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual.

In topology, Poincaré duality also reverses dimensions ; it corresponds to the fact that, if a topological manifold is respresented as a cell complex, then the dual of the complex ( a higher dimensional generalization of the planar graph dual ) represents the same manifold.

The map is then a graph, in which the nodes corresponds to places and arcs correspond to the paths.

Each node of the graph corresponds to a factor U ( N ) of the gauge group, and each link represents a field in the bifundamental representation

In the resulting graph, the source corresponds to the winner.

König's theorem in graph theory states that a minimum vertex cover in a bipartite graph corresponds to a maximum matching, and vice versa ; it can be interpreted as the perfection of the complements of bipartite graphs.

* A maximum independent set in a line graph corresponds to maximum matching in the original graph.