Alphabeta MathHow statement and proof provenance work
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The parameter kappa(G)=alpha(G)omega(G)
Definition
For a finite graph , define
where and are the stability number and clique number of (Cliques, stable sets, the clique number and stability number ).
In particular, because both factors vanish on the null graph.
Depends on
Used by
- Every perfect graph has a clique or stable set of size at least the square root of its order Corollary
- A tau-critical graph Definition
- A two-block pure blockade can realize equality in the additive kappa theorem Example
- A minimal counterexample to a kappa-bound is tau-critical Proposition
- A pure blockade with a cograph pattern has additive kappa Theorem
- A tau-critical graph has no wide pure blockade with cograph pattern Theorem
- Every perfect graph satisfies |V(G)|<=kappa(G) Theorem
- The Erdos-Hajnal property is equivalent to the large-cograph, large-perfect, and kappa formulations Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Maria Chudnovsky, Alex Scott, Paul Seymour, and Sophie Spirkl, Erdos-Hajnal for graphs with no 5-hole, Introduction (standard reference, not scraped)