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An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component. [6] An undirected graph can be decomposed into edge-disjoint cycles if and only if all of its vertices have even degree. So, a graph has an Eulerian cycle if and only if it ...
The vertices of odd degree in a graph are sometimes called odd nodes (or odd vertices); [4] in this terminology, the handshaking lemma can be rephrased as the statement that every graph has an even number of odd nodes.
The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, which is to prove that in any group of people, the number of people who have shaken hands with an odd ...
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 slightly more general case is a disconnected graph in which one or more components have an odd number of vertices (even if the total number of vertices is even). Let us call such components odd components. In any matching, each vertex can only be matched to vertices in the same component.
If G is a regular graph of degree d whose edge connectivity is at least d − 1, and G has an even number of vertices, then it has a perfect matching. More strongly, every edge of G belongs to at least one perfect matching.
Cycle graphs with an even number of vertices are bipartite. [4] Every planar graph whose faces all have even length is bipartite. [9] Special cases of this are grid graphs and squaregraphs, in which every inner face consists of 4 edges and every inner vertex has four or more neighbors. [10]
2. An odd vertex is a vertex whose degree is odd. By the handshaking lemma every finite undirected graph has an even number of odd vertices. 3. An odd ear is a simple path or simple cycle with an odd number of edges, used in odd ear decompositions of factor-critical graphs; see ear. 4.