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A forest complement and its star-expansion have the Erdős-Hajnal property
Statement
Let be a forest, and let be its star-expansion. Then the pair has the Erdős-Hajnal property.
Facts & Assumptions
Given: A forest .
Theorem 7.2 of the cited primary source proves exactly that has the Erdős-Hajnal property for every forest .
Proof
The cited primary-source theorem treats the complement-side low-degree case separately, using the forest-free sparse-pair theorem, and treats the graph-side case with the quantitative comb construction.
Its two cases exclude a critical counterexample and yield exactly the Erdős-Hajnal property for .
Depends on
- The star-expansion of a graph
- The star-expansion four-family of a forest has the Erdős-Hajnal property
- Every forest-free graph has a linear anticomplete pair or a linear-degree vertex
- A minimal counterexample to a kappa-bound is tau-critical
- An H-free graph has a linearly large induced subgraph whose graph or complement has bounded maximum degree
- The Erdos-Hajnal property is equivalent to the large-cograph, large-perfect, and kappa formulations
Used by
Dependency tree · two levels
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Sources
- Maria Chudnovsky, Alex Scott, Paul Seymour, and Sophie Spirkl, Erdős-Hajnal for graphs with no 5-hole, Theorem 7.2 (standard reference, not scraped)