In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
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, 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 the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs.
In the mathematical discipline of graph theory, a feedback vertex set (FVS) of a graph is a set of vertices whose removal leaves a graph without cycles. Equivalently, each FVS contains at least one vertex of any cycle in the graph. The feedback vertex set number of a graph is the size of a smallest feedback vertex set. The minimum feedback vertex set problem is an NP-complete problem; it was among the first problems shown to be NP-complete. It has wide applications in operating systems, database systems, and VLSI chip design.
In graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at least one edge out of every cycle in the graph. Removing these edges from the graph breaks all of the cycles, producing an acyclic subgraph of the given graph, often called a directed acyclic graph. A feedback arc set with the fewest possible edges is a minimum feedback arc set and its removal leaves a maximum acyclic subgraph; weighted versions of these optimization problems are also used. If a feedback arc set is minimal, meaning that removing any edge from it produces a subset that is not a feedback arc set, then it has an additional property: reversing all of its edges, rather than removing them, produces a directed acyclic graph.
The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph. The Nash-Williams theorem provides necessary and sufficient conditions for when a graph is k-arboric.
The circle packing theorem describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles whose interiors are disjoint. The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph; more generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs or contact graphs. Coin graphs are always connected, simple, and planar. The circle packing theorem states that these are the only requirements for a graph to be a coin graph:
In graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge that makes it into a strongly connected graph.
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 linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two; that is, it is a disjoint union of path graphs. Linear arboricity is a variant of arboricity, the minimum number of forests into which the edges can be partitioned.
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.
Alspach's conjecture is a mathematical theorem that characterizes the disjoint cycle covers of complete graphs with prescribed cycle lengths. It is named after Brian Alspach, who posed it as a research problem in 1981. A proof was published by Darryn Bryant, Daniel Horsley, and William Pettersson.
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph.
In mathematics, a dicut is a partition of the vertices of a directed graph into two subsets, so that each edge that has an endpoint in both subsets is directed from the first subset to the second. Each strongly connected component of the graph must be entirely contained in one of the two subsets, so a strongly connected graph has no nontrivial dicuts.
In the mathematics of directed graphs, the Lucchesi–Younger theorem is a relationship between dicuts and dijoins. It was published by Cláudio L. Lucchesi and Daniel H. Younger in 1978. Their proof resolved a conjecture that had been posed roughly a decade earlier by Younger, and in unpublished work by Neil Robertson, motivated by the duality in planar graphs between dijoins and feedback arc sets.
In mathematics, a dijoin is a subset of the edges of a directed graph, with the property that contracting every edge in the dijoin produces a strongly connected graph. Equivalently, a dijoin is a subset of the edges that, for every dicut, includes at least one edge crossing the dicut. Here, a dicut is a partition of the vertices into two subsets, so that each edge that has an endpoint in both subsets is directed from the first subset to the second.
In the theory of combinatorial optimization, submodular flow is a general class of optimization problems that includes as special cases the minimum-cost flow problem, matroid intersection, and the problem of computing a minimum-weight dijoin in a weighted directed graph. It was originally formulated by Jack Edmonds and Rick Giles, and can be solved in polynomial time.
In graph theory, the Lovász–Woodall conjecture is a long-standing problem on cycles in graphs. It says:
