Cycle graph
A cycle graph of length 6
Vertices	n
Edges	n
Chromatic number	3 if n is odd 2 if n is even
Properties	2-regular Unit distance

In graph theory, a cycle graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called C_n. The number of vertices in a C_n equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. A cycle graph or cyclic graph is a connected, 2-regular graph. ... Image File history File links Undirected_6_cycle. ... This article just presents the basic definitions. ... This article just presents the basic definitions. ... A 3_coloring suits this graph, but fewer colors would result in adjacent verticies of the same color. ... In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i. ... The Petersen graph is a unit distance graph: it can be drawn in the plane with each edge having unit length. ... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. ... In graph theory, the degree (or valency) of a vertex is the number of edges incident to the vertex. ...

A note on terminology

There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or n-gon are also often used. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Synonyms (in ancient Greek, συν (syn) = plus and όνομα (onoma) = name) are different words with similar or identical meanings. ... In mathematics and computer science, graph theory studies the properties of graphs. ...

Properties

A cycle graph is:

• connected
• 2-regular
• Eulerian
• Hamiltonian
• 2-vertex colorable, if and only if it has an even number of vertices. More generally, a graph is bipartite if and only if it has no odd cycles (Kőnig, 1936).
• 2-edge colorable, if and only if it has an even number of vertices
• 3-vertex colorable and 3-edge colorable, for any number of vertices
• A unit distance graph

In addition: In mathematics and computer science, graph theory studies the properties of graphs. ... In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i. ... Bridges of Königsberg corresponding graph In the mathematical field of graph theory an Eulerian path is a path in a graph which visits each edge exactly once. ... In the mathematical field of graph theory, a Hamiltonian path is a path in a undirected graph which visits each vertex exactly once. ... In the mathematical field of graph theory, a bipartite graph is a special graph where the set of vertices can be divided into two disjunct sets with two vertices of the same set never sharing an edge. ... It has been suggested that this article or section be merged into Logical biconditional. ... Dénes KÅ‘nig Dénes KÅ‘nig, (September 21, 1884 – October 19, 1944), was a Hungarian mathematician who worked in the field of graph theory. ... 3-edge-coloring of Desargues graph. ... The Petersen graph is a unit distance graph: it can be drawn in the plane with each edge having unit length. ...

• As cycle graphs can be drawn as regular polygons, the symmetries of an n-cycle are the same as those of a regular polygon with n sides, the dihedral group of order 2n. In particular, there exist symmetries taking any vertex to any other vertex, and any edge to any other edge, so the n-cycle is a symmetric graph.

As a branch of Graph theory, Graph drawing applies topology and geometry to derive visual and haptic representations of graphs. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... This article may be confusing for some readers, and should be edited to enhance clarity. ... In graph theory a graph is symmetric or arc transitive if it is both vertex transitive and edge transitive Categories: Algebraic graph theory | Graph theory ...

Directed cycle graph

A directed cycle graph of length 8

A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. Image File history File links No higher resolution available. ...

In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a feedback vertex set. In task scheduling, the edges may represent precedence scheduling constraints, and we need to identify the smallest number of constraints that must be dropped so as to permit a valid schedule; namely, the graph of tasks and constraints does not contain directed cycles. ... BEAT YOU ...

A directed cycle graph has uniform in-degree 1 and uniform out-degree 1.

Directed cycle graphs are Cayley graphs for cyclic groups (see e.g. Trevisan). The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

External links

• Eric W. Weisstein, Cycle Graph at MathWorld. (MathWorld discusses both 2-regular cycle graphs and the group-theoretic concept of cycle diagrams in the same article.)

• Luca Trevisan, Characters and Expansion.