Math 2.0

statement theorem

Every connected graph has a spanning 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.

Explanations

L1 human

If a connected graph has a loop, you can remove one edge of the loop and the graph stays in one piece. Repeat until there are no loops. What is left is a spanning tree.

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What this needs

To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

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 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. 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

Sources

  1. Web https://en.wikipedia.org/wiki/Spanning_tree, Definitions

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