Math 2.0

statement proposition

Chromatic number of a complete bipartite graph

Statement

Let m,n≥1m, n \ge 1 be integers. Then the chromatic number of the complete bipartite graph Km,nK_{m,n} is 22.

Remarks

Give one colour to each of the two parts. Each edge joins the two parts, so this is a colouring with 22 colours. One colour is not enough, because Km,nK_{m,n} has an edge. More generally, a bipartite graph with at least one edge has chromatic number 22.

Explanations

L1 human

Colour one side of Km,nK_{m,n} red and the other side blue. Every edge goes from red to blue, so two colours are enough, and one colour is too few.

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Complete bipartite graph, example, no evidence levelComplete bipartite gr…exampleGraph colouring, definition, no evidence levelGraph colouringdefinitionK-colourable graph, definition, no evidence levelK-colourable graphdefinitionChromatic number, definition, no evidence levelChromatic numberdefinitionChromatic number of a complete bipartite graph, statement, no evidence levelChromatic number of a…statement
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  1. Complete bipartite graph example
  2. Graph colouring definition
  3. K-colourable graph definition
  4. Chromatic number definition
  5. Chromatic number of a complete bipartite graph statement

Sources

  1. Web https://en.wikipedia.org/wiki/Complete_bipartite_graph, lead section

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