Search results
Results From The WOW.Com Content Network
Any complete graph is a core. A cycle of odd length is a core. A graph is a core if and only if the core of is equal to . Every two cycles of even length, and more generally every two bipartite graphs are hom-equivalent. The core of each of these graphs is the two-vertex complete graph K 2.
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one 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.
From the handshaking lemma, a k-regular graph with odd k has an even number of vertices. A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2]
A complete graph with n nodes represents the edges of an (n – 1)-simplex. Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton. [15] Every neighborly polytope in four or more dimensions also has a ...
In some variations of this problem, the output should list all cliques of size k. [18] In the clique decision problem, the input is an undirected graph and a number k, and the output is a Boolean value: true if the graph contains a k-clique, and false otherwise. [19] The first four of these problems are all important in practical applications.
The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f ( v ) to each vertex v , a graph G is f -choosable (or f -list-colorable ) if it has a list coloring no matter how one assigns a list of f ( v ...
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!
A claw is a tree with one internal vertex and three leaves, or equivalently the complete bipartite graph K 1,3. A claw-free graph is a graph that does not have an induced subgraph that is a claw. clique A clique is a set of mutually adjacent vertices (or the complete subgraph induced by that set). Sometimes a clique is defined as a maximal set ...