CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The star-expansion of the four-vertex path and its complement have the Erdős-Hajnal property

Statement

Let $P_{4}$ be the four-vertex path, and let $P_{4}^{⋆}$ be its star-expansion. Then ${P_{4}^{⋆}, \overline{P_{4}^{⋆}}}$ has the Erdős-Hajnal property.

Facts & Assumptions

Given: The four-vertex path $P_{4}$ .

[L1]

For every forest $F$ , the four graphs $F^{⋆}, (\overline{F})^{⋆}, \overline{F^{⋆}}, \overline{(\overline{F})^{⋆}}$ have the Erdős-Hajnal property as a family (The star-expansion four-family of a forest has the Erdős-Hajnal property).

[F1]

The path $P_{4}$ is self-complementary: if its vertices in order are $1, 2, 3, 4$ , then the bijection $1 \mapsto 2$ , $2 \mapsto 4$ , $3 \mapsto 1$ , $4 \mapsto 3$ identifies $P_{4}$ with $\overline{P_{4}}$ .

Proof

technique · direct

1.1

Apply [L1] with $F : = P_{4}$ . Because $P_{4}$ is a forest, the family ${P_{4}^{⋆}, (\overline{P_{4}})^{⋆}, \overline{P_{4}^{⋆}}, \overline{(\overline{P_{4}})^{⋆}}}$ has the Erdős-Hajnal property.

2.1

By [F1], $P_{4} ≅ \overline{P_{4}}$ , so $(\overline{P_{4}})^{⋆} ≅ P_{4}^{⋆}$ and $\overline{(\overline{P_{4}})^{⋆}} ≅ \overline{P_{4}^{⋆}}$ . Hence the four-family in step 1.1 collapses to the two-family ${P_{4}^{⋆}, \overline{P_{4}^{⋆}}}$ .

step 1.1F1

3.1

Therefore the star-expansion of $P_{4}$ and its complement have the Erdős-Hajnal property.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

16 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, Erdős-Hajnal for graphs with no 5-hole, Theorem 6.2 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Theorem 1.9 discussion (standard reference, not scraped)