Math 2.0

Combinatorics Theorem

Every connected graph has a spanning tree

Builds on Connected graphSpanning tree

Statement

Let GG be a connected graph. Then GG has a spanning tree.

Remarks

For a finite graph, a proof deletes edges that lie on cycles, one at a time. Such an edge is not a bridge, so the graph stays connected, and the process ends with a tree. For an infinite graph, the proof uses Zorn's lemma, or a well-ordering of the vertices. The converse also holds: a graph with a spanning tree is connected.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionSubgraph, definition, no evidence levelSubgraphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, definition, no evidence levelReachabilitydefinitionTrail, definition, no evidence levelTraildefinitionConnected graph, definition, no evidence levelConnected graphdefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionAcyclic graph, definition, no evidence levelAcyclic graphdefinitionTree, definition, no evidence levelTreedefinitionSpanning tree, definition, no evidence levelSpanning treedefinitionEvery connected graph has a spanning tree, statement, no evidence levelEvery connected graph…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. Graph definition
  2. Subgraph definition
  3. Walk definition
  4. Reachability definition
  5. Trail definition
  6. Connected graph definition
  7. Circuit definition
  8. Cycle definition
  9. Acyclic graph definition
  10. Tree definition
  11. Spanning tree definition
  12. Every connected graph has a spanning tree statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Spanning_tree, Definitions
About this page
Tag04XPQT2XTC
URIm2://04XPQT2XTC
Kindstatement, theorem
WritingW0 stub
MSC05C05
StatusThis copy of the site has no build data. Run m2 status 04XPQT2XTC.
UsesConnected graph, Spanning tree

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