In graph theory, a critical graph is an undirected graph all of whose proper subgraphs have smaller chromatic number. In such a graph, every vertex or edge is a critical element, in the sense that its deletion would decrease the number of colors needed in a graph coloring of the given graph. The decrease in the number of colors cannot be by more than one.
A -critical graph is a critical graph with chromatic number . A graph with chromatic number is -vertex-critical if each of its vertices is a critical element. Critical graphs are the minimal members in terms of chromatic number, which is a very important measure in graph theory.
Some properties of a -critical graph with vertices and edges:
Graph is vertex-critical if and only if for every vertex , there is an optimal proper coloring in which is a singleton color class.
As Hajós (1961) showed, every -critical graph may be formed from a complete graph by combining the Hajós construction with an operation that identifies two non-adjacent vertices. The graphs formed in this way always require colors in any proper coloring. [8]
A double-critical graph is a connected graph in which the deletion of any pair of adjacent vertices decreases the chromatic number by two. It is an open problem to determine whether is the only double-critical -chromatic graph. [9]
The Turán graph, denoted by , is a complete multipartite graph; it is formed by partitioning a set of vertices into subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to different subsets. Where and are the quotient and remainder of dividing by , the graph is of the form , and the number of edges is
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.
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various graph properties, both global and local, and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory.
In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices.
In graph theory, the perfect graph theorem of László Lovász states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by Berge, and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theorem characterizing perfect graphs by their forbidden induced subgraphs.
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, the degree of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex is denoted or . The maximum degree of a graph , denoted by , and the minimum degree of a graph, denoted by , are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0.
In graph theory, the Hadwiger conjecture states that if is loopless and has no minor then its chromatic number satisfies . It is known to be true for . The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field.
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 graph theory, a mathematical discipline, a factor-critical graph is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching.
In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges an -vertex graph can have such that it does not have a subgraph isomorphic to . In this context, is called a forbidden subgraph.
András Hajnal was a professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics.
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 graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with colors, the same is true for the whole graph. The theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős (1951), after whom it is named.
In the mathematical field of graph theory, the friendship graphFn is a planar, undirected graph with 2n + 1 vertices and 3n edges.
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
In graph theory, a branch of mathematics, the Hajós construction is an operation on graphs named after György Hajós (1961) that may be used to construct any critical graph or any graph whose chromatic number is at least some given threshold.
In graph theory, the Gallai–Hasse–Roy–Vitaver theorem is a form of duality between the colorings of the vertices of a given undirected graph and the orientations of its edges. It states that the minimum number of colors needed to properly color any graph equals one plus the length of a longest path in an orientation of chosen to minimize this path's length. The orientations for which the longest path has minimum length always include at least one acyclic orientation.
In graph theory, a branch of mathematics, the Erdős–Hajnal conjecture states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent sets. It is named for Paul Erdős and András Hajnal.
The Earth–Moon problem is an unsolved problem on graph coloring in mathematics. It is an extension of the planar map coloring problem, and was posed by Gerhard Ringel in 1959. In mathematical terms, it seeks the chromatic number of biplanar graphs. It is known that this number is at least 9 and at most 12.
