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If path ends at vertex , then the vertex corresponding to in has degree equal to the number of ways that may be extended by an edge that does not connect back to ; that is, the degree of this vertex in is either (an even number) if does not form part of a Hamiltonian cycle through , or (an odd number) if is part of a Hamiltonian cycle ...
A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]
The odd graph has one vertex for each of the ()-element subsets of a ()-element set. Two vertices are connected by an edge if and only if the corresponding subsets are disjoint . [ 2 ] That is, O n {\displaystyle O_{n}} is the Kneser graph K G ( 2 n − 1 , n − 1 ) {\displaystyle KG(2n-1,n-1)} .
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 ...
A cycle graph is: 2-edge colorable, if and only if it has an even number of vertices; 2-regular; 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. 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.
A classic result of graph theory states that a graph of odd order (having an odd number of vertices) always has at least one vertex of even degree. (The statement itself requires zero to be even: the empty graph has an even order, and an isolated vertex has an even degree.) [ 16 ] In order to prove the statement, it is actually easier to prove ...
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. Therefore, any matching leaves at least one unmatched ...