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For every $k$ , the class forbidding $P_{k}$ and $\overline{P_{k}}$ has the Erdős–Hajnal property

Statement

For every integer $k \geq 2$ , every finite graph with no induced $P_{k}$ and no induced $\overline{P_{k}}$ has a clique or stable set of size at least a positive power of its order. Equivalently, the class forbidding $P_{k}$ and $\overline{P_{k}}$ has the Erdős–Hajnal property.

Facts & Assumptions

Given: An integer $k \geq 2$ .

[L1]

For this $k$ , the class forbidding $P_{k}$ and $\overline{P_{k}}$ has the strong Erdős–Hajnal property (For every $k$ , the class forbidding $P_{k}$ and $\overline{P_{k}}$ has the strong Erdős–Hajnal property).

[L2]

Every hereditary class with the strong Erdős–Hajnal property has the Erdős–Hajnal property (The strong Erdős–Hajnal property implies the Erdős–Hajnal property).

Proof

technique · direct

1.1

The previous theorem gives the strong Erdős–Hajnal property for the hereditary class of graphs forbidding $P_{k}$ and $\overline{P_{k}}$ .

2.1

Applying [L2] to that class yields the Erdős–Hajnal property.

step 1.1L2∎

Depends on

Used by

The path-antipath theorem specialized to P₅ and P₅ Example

Dependency tree · two levels

18 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

Nicolas Bousquet, Aurélie Lagoutte, and Stéphan Thomassé, The Erdős-Hajnal Conjecture for Paths and Antipaths, Theorem 4 (standard reference, not scraped)

For every kk, the class forbidding PkPk​ and Pk‾Pk​​ has the Erdős–Hajnal property