Search results
Results From The WOW.Com Content Network
Let G be a k-regular graph with 2n nodes. If k is sufficiently large, it is known that G has to be 1-factorable: If k = 2n − 1, then G is the complete graph K 2n, and hence 1-factorable (see above). If k = 2n − 2, then G can be constructed by removing a perfect matching from K 2n. Again, 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.
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).
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
Let G be a simple plane graph with n vertices; we may add edges if necessary so that G is a maximally plane graph. If n < 3, the result is trivial. If n ≥ 3, then all faces of G must be triangles, as we could add an edge into any face with more sides while preserving planarity, contradicting the assumption of maximal planarity.
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 ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
Next, consider a graph G with a vertex u such that, if we remove from G the vertex u and its adjacent edges, the remaining graph (denoted G − u) has two or more odd components. As above, any matching leaves, in every odd component, at least one vertex that is unmatched to other vertices in the same component.