Math 2.0

statement theorem

A graph is bipartite iff it has no odd cycle

Statement

Let GG be a graph. Then GG is bipartite if and only if GG has no cycle of odd length.

Remarks

One direction is easy. A cycle in a bipartite graph alternates between the two parts, so its length is even. For the converse, work in each component. Colour a vertex by the parity of its distance to a fixed vertex. An edge between two vertices of the same colour would give an odd closed walk, and then an odd cycle. A graph is also bipartite if and only if its chromatic number is at most 22.

Explanations

L1 human

Walking around a cycle in a bipartite graph, you switch sides at every step. To come back to the start you need an even number of steps, so odd cycles are impossible.

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Bipartite graph, definition, no evidence levelBipartite graphdefinitionGraph, definition, no evidence levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionTrail, definition, no evidence levelTraildefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionA graph is bipartite iff it has no odd cycle, statement, no evidence levelA graph is bipartite …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. Bipartite graph definition
  2. Graph definition
  3. Walk definition
  4. Trail definition
  5. Circuit definition
  6. Cycle definition
  7. A graph is bipartite iff it has no odd cycle statement

Sources

  1. Web https://en.wikipedia.org/wiki/Bipartite_graph, Characterization

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