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The Erdős–Gallai theorem is a result in graph theory, a branch of combinatorial mathematics.It provides one of two known approaches to solving the graph realization problem, i.e. it gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph.
Again, G is 1-factorable. Chetwynd & Hilton (1985) show that if k ≥ 12n/7, then G is 1-factorable. The 1-factorization conjecture [3] is a long-standing conjecture that states that k ≈ n is sufficient. In precise terms, the conjecture is: If n is odd and k ≥ n, then G is 1-factorable. If n is even and k ≥ n − 1 then G is 1-factorable.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Then by the Tutte theorem G contains a perfect matching. Let G i be a component with an odd number of vertices in the graph induced by the vertex set V − U. Let V i denote the vertices of G i and let m i denote the number of edges of G with one vertex in V i and one vertex in U. By a simple double counting argument we have that
Maximum cardinality matching is a fundamental problem in graph theory. [1] We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this ...
[1] [2] A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph. [3] This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily ...
Theorem [7] [8] — A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed. The theorem is a consequence of the open mapping theorem ; see § Relation to the open mapping theorem below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).
H(n)/G(n) goes to 0 as n goes to infinity exponentially rapidly, where H(n) is the number of (non-isomorphic) highly irregular graphs with n vertices, and G(n) is the total number of graphs with n vertices. [3] For every graph G, there exists a highly irregular graph H containing G as an induced subgraph. [3] This last observation can be ...