In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.
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, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . A set is independent if and only if it is a clique in the graph’s complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening.
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 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 graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic number A(G) of a graph G is the fewest colors needed in any acyclic coloring of G.
In graph theory, a circle graph is the intersection graph of a set of chords of a circle. That is, it is an undirected graph whose vertices can be associated with chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane. It was proven by Jeremie Chalopin and Daniel Gonçalves (2009).
In graph theory, the Grundy number or Grundy chromatic number of an undirected graph is the maximum number of colors that can be used by a greedy coloring strategy that considers the vertices of the graph in sequence and assigns each vertex its first available color, using a vertex ordering chosen to use as many colors as possible. Grundy numbers are named after P. M. Grundy, who studied an analogous concept for directed graphs in 1939. The undirected version was introduced by Christen & Selkow (1979).
In graph theory, boxicity is a graph invariant, introduced by Fred S. Roberts in 1969.
In graph theory, a string graph is an intersection graph of curves in the plane; each curve is called a "string". Given a graph G, G is a string graph if and only if there exists a set of curves, or strings, drawn in the plane such that no three strings intersect at a single point and such that the graph having a vertex for each curve and an edge for each intersecting pair of curves is isomorphic to G.
In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices.
In graph-theoretic mathematics, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by Grünbaum (1973). The star chromatic number of G is the fewest colors needed to star color G.
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not in general use the minimum number of colors possible.
In graph theory, a perfectly orderable graph is a graph whose vertices can be ordered in such a way that a greedy coloring algorithm with that ordering optimally colors every induced subgraph of the given graph. Perfectly orderable graphs form a special case of the perfect graphs, and they include the chordal graphs, comparability graphs, and distance-hereditary graphs. However, testing whether a graph is perfectly orderable is NP-complete.
In graph theory, a mathematical discipline, coloring refers to an assignment of colours or labels to vertices, edges and faces of a graph. Defective coloring is a variant of proper vertex coloring. In a proper vertex coloring, the vertices are coloured such that no adjacent vertices have the same colour. In defective coloring, on the other hand, vertices are allowed to have neighbours of the same colour to a certain extent.
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, a branch of mathematics, the kth powerGk of an undirected graph G is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in G is at most k. Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: G2 is called the square of G, G3 is called the cube of G, etc.
In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a proper coloring: adjacent vertices are allowed to be given the same color. For the colored graph, there should not exist any one-to-one mapping of the vertices to themselves that preserves both adjacency and coloring. The minimum number of colors in a distinguishing coloring is called the distinguishing number of the graph.
In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers.
