Math 2.0

statement corollary

Number of odd-degree vertices is even

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.

Explanations

L1 human

The total of all degrees is even, because it is twice the number of edges. A sum of whole numbers is even only if it has an even number of odd terms.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Handshaking_lemma, Definitions and statement

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