Clique cover number θ(G): minimum cliques partitioning all vertices. θ(G) = χ(Ḡ) — coloring the complement. θ(K_n)=1 (one clique). θ(E_n)=n (n isolated vertices = n trivial cliques). Perfect graphs: θ=α (clique cover = independence number).
How to Use
Select graph:
- θ: Clique cover number
- Dual: θ=χ(Ḡ)
- Perfect: θ=α
Complement Duality
θ(G) = χ(Ḡ): covering G with cliques = coloring Ḡ. Cliques in G are independent sets in Ḡ. This duality is fundamental to perfect graph theory (Lovász, 1972).
Perfect Graphs
For perfect graphs: θ(G)=α(G) for all induced subgraphs. The Strong Perfect Graph Theorem (Chudnovsky et al. 2006): G is perfect iff it has no odd hole or odd antihole. This settles Berge's 1961 conjecture.
Step-by-Step Instructions
- 1Select graph.
- 2Compute θ(G).
- 3Compare to χ(Ḡ).
- 4Check perfection.
- 5Find cover.