Cycle spaceIn graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings.
Petersen familyIn graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph K_6. The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph. Any of the graphs in the Petersen family can be transformed into any other graph in the family by Δ-Y or Y-Δ transforms, operations in which a triangle is replaced by a degree-three vertex or vice versa.
Graph factorizationIn graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors. A 2-factor is a collection of cycles that spans all vertices of the graph.
Moore graphIn graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). If the degree of such a graph is d and its diameter is k, its girth must equal 2k + 1. This is true, for a graph of degree d and diameter k, if and only if its number of vertices equals an upper bound on the largest possible number of vertices in any graph with this degree and diameter. Therefore, these graphs solve the degree diameter problem for their parameters.
Fractional coloringFractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a set of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common.
Linkless embeddingIn topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs.
Cage (graph theory)In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an (r, g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g. An (r, g)-cage is an (r, g)-graph with the smallest possible number of vertices, among all (r, g)-graphs. A (3, g)-cage is often called a g-cage. It is known that an (r, g)-graph exists for any combination of r ≥ 2 and g ≥ 3.
Tutte polynomialThe Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph and contains information about how the graph is connected. It is denoted by . The importance of this polynomial stems from the information it contains about .
Hadwiger conjecture (graph theory)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 more detail, if all proper colorings of an undirected graph use or more colors, then one can find disjoint connected subgraphs of such that each subgraph is connected by an edge to each other subgraph.
Degree diameter problemIn graph theory, the degree diameter problem is the problem of finding the largest possible graph G (in terms of the size of its vertex set V) of diameter k such that the largest degree of any of the vertices in G is at most d. The size of G is bounded above by the Moore bound; for 1 < k and 2 < d only the Petersen graph, the Hoffman-Singleton graph, and possibly one more graph (not yet proven to exist) of diameter k = 2 and degree d = 57 attain the Moore bound.
