Graceful labeling: assign distinct labels 0..m to vertices of graph with m edges, so edge differences |f(u)−f(v)| give {1,2,...,m}. The Ringel-Kotzig conjecture (1967): every tree is graceful. Verified for trees with ≤35 vertices. Known graceful: paths, stars, caterpillars, spiders, lobsters.
How to Use
Define a graph by edges:
- Labeling: Find graceful assignment
- Edge diffs: Verify completeness
- Type: Path/star/cycle/tree
The Conjecture
Ringel (1964) and Kotzig (1967): all trees are graceful. Unproven after 60 years! Equivalent to decomposing K_{2m+1} into copies of any tree T with m edges. Verified computationally for small trees.
Graceful Graphs
- Paths Pₙ: always (label: 0,m,1,m-1,...)
- Stars K_{1,m}: always (center=0, leaves=1..m)
- Cycles Cₙ: iff n≡0,3 (mod 4)
- Complete bipartite K_{m,n}: always
Step-by-Step Instructions
- 1Define graph edges.
- 2Attempt labeling.
- 3Check edge diffs.
- 4Verify coverage.
- 5Identify type.