Math 2.0

Combinatorics Planar graphs, embeddings, minors Theorem

Kruskal's tree theorem

Builds on Rooted tree

Statement

For finite rooted trees TT and T′T', write u∧vu \wedge v for the nearest common ancestor of vertices u,vu, v of TT. Say that TT embeds in T′T' if some injective map ff from the vertices of TT to those of T′T' satisfies f(u∧v)=f(u)∧f(v)f(u \wedge v) = f(u) \wedge f(v) for all u,vu, v. Then the finite rooted trees are well-quasi-ordered by embedding. That is, for every infinite sequence T1,T2,T3,…T_1, T_2, T_3, \dots of finite rooted trees, there are indices i<ji < j such that TiT_i embeds in TjT_j.

Remarks

The labelled form also holds. Let the vertices carry labels from a well-quasi-ordered set (X,≤)(X, \le), and ask also that each label of TT is at most the label of its image. Then the trees are still well-quasi-ordered. Joseph Kruskal proved the theorem in 1960, and Crispin Nash-Williams gave a short proof by the minimal bad sequence argument. That argument takes a bad sequence whose trees are as small as possible at each step. It then applies Higman's lemma to the sets of subtrees at the roots to reach a contradiction. Harvey Friedman showed that a finite form of the theorem, although true, is not provable in the subsystem ATR0\mathrm{ATR}_0 of second-order arithmetic. The Robertson–Seymour theorem on graph minors builds on its methods.

Going deeper

For a student

The theorem is a basic example of a well-quasi-order on structured objects, after Higman's lemma for finite words. It is used in term rewriting to prove that certain orderings, such as recursive path orderings, are well-founded. Its finite form grows so fast that it escapes strong formal systems, which makes it a standard example in proof theory.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionWalk, 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 levelTreedefinitionRooted tree, definition, no evidence levelRooted treedefinitionKruskal's tree theorem, statement, no evidence levelKruskal's tree theoremstatement
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. Walk definition
  3. Reachability definition
  4. Trail definition
  5. Connected graph definition
  6. Circuit definition
  7. Cycle definition
  8. Acyclic graph definition
  9. Tree definition
  10. Rooted tree definition
  11. Kruskal's tree theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Kruskal's_tree_theorem
About this page
TagXVHKCX4F5Y
URIm2://XVHKCX4F5Y
Kindstatement, theorem
WritingW0 stub
MSC05C05, 06A07
StatusThis copy of the site has no build data. Run m2 status XVHKCX4F5Y.
AboutRooted tree
UsesRooted tree
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.