Math 2.0

Combinatorics Theorem

A graph is bipartite iff it has no odd cycle

Builds on CycleBipartite graph

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.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
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 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. 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

Further reading

  1. Web https://en.wikipedia.org/wiki/Bipartite_graph, Characterization
About this page
Tag01R1HPEK6S
URIm2://01R1HPEK6S
Kindstatement, theorem
WritingW0 stub
MSC05C75
StatusThis copy of the site has no build data. Run m2 status 01R1HPEK6S.
UsesCycle, Bipartite graph

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