Math 2.0

statement proposition

K5 and K3,3 are not planar

Statement

The complete graph K5K_5 and the complete bipartite graph K3,3K_{3,3} are not planar.

Remarks

The graph K5K_5 has 55 vertices and 1010 edges. The edge bound for planar graphs allows at most 3⋅5−6=93 \cdot 5 - 6 = 9 edges. The graph K3,3K_{3,3} has 66 vertices and 99 edges, and it has no cycle of length 33. The bound e≤2v−4e \le 2v - 4 allows at most 88 edges. The statement for K3,3K_{3,3} is the three utilities problem: three houses cannot each be joined to three utilities without crossings.

Explanations

L1 human

Try to join each of three houses to each of three utilities with lines that never cross. It is impossible, and the reason is a count of edges and faces.

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Complete bipartite graph, example, no evidence levelComplete bipartite gr…exampleComplete graph, example, no evidence levelComplete graphexampleGraph, definition, no evidence levelGraphdefinitionPlanar graph, definition, no evidence levelPlanar graphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, definition, no evidence levelReachabilitydefinitionConnected graph, definition, no evidence levelConnected graphdefinitionEuler's formula for planar graphs, statement, no evidence levelEuler's formula for p…statementEdge bound for planar graphs, statement, no evidence levelEdge bound for planar…statementK5 and K3,3 are not planar, statement, no evidence levelK5 and K3,3 are not p…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.
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  1. Complete bipartite graph example
  2. Complete graph example
  3. Graph definition
  4. Planar graph definition
  5. Walk definition
  6. Reachability definition
  7. Connected graph definition
  8. Euler's formula for planar graphs statement
  9. Edge bound for planar graphs statement
  10. K5 and K3,3 are not planar statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Three_utilities_problem, Unsolvability

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