Graph power G^k: uv adjacent iff d_G(u,v)≤k. G² (square): distance ≤ 2. G³ (cube): distance ≤ 3. diam(G^k) = ⌈diam(G)/k⌉. Fleischner (1974): if G is 2-connected, then G² is Hamiltonian.
How to Use
Select graph and k:
- G^k: Power graph
- Edges: Distance ≤ k
- Diameter: Reduced
Fleischner's Theorem
Every 2-connected graph G has a Hamiltonian square G². This means the square always has a Hamiltonian cycle. The proof is constructive but complex. The theorem doesn't extend to cubes of 1-connected graphs.
Applications
Network design: G^k represents k-hop communication. Frequency assignment: G² models 2-step interference. Graph coloring: χ(G^k) bounds relate to distance coloring. Distributed computing: k-step message passing.
Step-by-Step Instructions
- 1Select graph.
- 2Choose k.
- 3Compute G^k.
- 4Count new edges.
- 5Check diameter.