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The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. K n can be decomposed into n trees T i such that T i has i vertices. [6] Ringel's conjecture asks if the complete graph K 2n+1 can be decomposed into copies of any tree ...
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 complement graph ¯ of a simple graph G is another graph on the same vertex set as G, with an edge for each two vertices that are not adjacent in G. complete 1. A complete graph is one in which every two vertices are adjacent: all edges that could exist are present. A complete graph with n vertices is often denoted K n.
A graph is planar if it contains as a subdivision neither the complete bipartite graph K 3,3 nor the complete graph K 5. Another problem in subdivision containment is the Kelmans–Seymour conjecture: Every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph K 5.
As complete graphs are Hamiltonian, all graphs whose closure is complete are Hamiltonian, which is the content of the following earlier theorems by Dirac and Ore. Dirac's Theorem (1952) — A simple graph with n vertices ( n ≥ 3 {\displaystyle n\geq 3} ) is Hamiltonian if every vertex has degree n 2 {\displaystyle {\tfrac {n}{2}}} or greater.
Therefore, the simplex graph contains within it a subdivision of G itself. The simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph. The simplex graph of the complement graph of a path graph is a Fibonacci cube.
Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph.
A complete graph (denoted , where is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, . In a signed graph , the number of positive edges connected to the vertex v {\displaystyle v} is called positive deg ( v ) {\displaystyle (v)} and the number of connected negative ...