Math 2.0

Combinatorics Corollary

Number of odd-degree vertices is even

Builds on Degree of a vertexHandshaking lemma

Statement

Let GG be a finite graph. Then the number of vertices of GG with odd degree is even.

Remarks

It follows from the handshaking lemma: the sum of all degrees is even, so it has an even number of odd terms. Thus no graph has exactly one vertex of odd degree. For example, no graph has the degrees 3,2,2,23, 2, 2, 2.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionEdge set, definition, no evidence levelEdge setdefinitionNeighbourhood, definition, no evidence levelNeighbourhooddefinitionDegree of a vertex, definition, no evidence levelDegree of a vertexdefinitionHandshaking lemma, statement, no evidence levelHandshaking lemmastatementNumber of odd-degree vertices is even, statement, no evidence levelNumber of odd-degree …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Edge set definition
  3. Neighbourhood definition
  4. Degree of a vertex definition
  5. Handshaking lemma statement
  6. Number of odd-degree vertices is even statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Handshaking_lemma, Definitions and statement
About this page
Tag0QQQQCPZA8
URIm2://0QQQQCPZA8
Kindstatement, corollary
WritingW0 stub
MSC05C07
StatusThis copy of the site has no build data. Run m2 status 0QQQQCPZA8.
UsesDegree of a vertex, Handshaking lemma

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