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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.
Dependent random choice can help find the Turán number.Using appropriate parameters, if = is a bipartite graph in which all vertices in have degree at most , then the extremal number (,) / where = only depends on .
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.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
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.
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).
For a graph , the deck of G, denoted (), is the multiset of isomorphism classes of all vertex-deleted subgraphs of . Each graph in () is called a card. Two graphs that have the same deck are said to be hypomorphic. With these definitions, the conjecture can be stated as:
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)