Cartesian product G□H: (u₁,v₁)~(u₂,v₂) iff (u₁=u₂ and v₁v₂∈E(H)) or (v₁=v₂ and u₁u₂∈E(G)). P_m□P_n = grid. K₂□K₂□...□K₂ = hypercube Q_n. C_m□C_n = torus. Commutative, associative.
How to Use
Select G and H:
- |V|: |V₁|·|V₂|
- |E|: Formula
- Type: Grid/torus/cube
Famous Products
P_m□P_n = m×n grid graph. C_m□C_n = torus graph. K₂□K₂ = C₄ (4-cycle). Q_n = K₂^□n = n-dimensional hypercube. The Petersen graph is the Kneser graph K(5,2), not a Cartesian product.
Properties
Commutative and associative. χ(G□H) = max(χ(G),χ(H)). κ(G□H) = κ(G)+κ(H) (connectivity adds). Hamiltonian if one factor is Hamiltonian and the other is connected.
Step-by-Step Instructions
- 1Select G, H.
- 2Compute G□H.
- 3Count vertices/edges.
- 4Identify structure.
- 5Check properties.