CorollaryStatement: AI-generatedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Substituting a complete or an edgeless graph for a vertex preserves the Erdős–Hajnal property

Statement

Let $H$ be a graph with the Erdős–Hajnal property and let $v \in V (H)$ . If $t \geq 1$ , then the graph obtained by substituting $K_{t}$ or $\overline{K_{t}}$ for $v$ also has the Erdős–Hajnal property.

Facts & Assumptions

Given: A graph $H$ with the Erdős–Hajnal property, a vertex $v \in V (H)$ , and an integer $t \geq 1$ .

[L1]

For every $t \geq 1$ , the class of $K_{t}$ -free graphs has the Erdős–Hajnal property (For every $t \geq 1$ , the class of $K_{t}$ -free graphs has the Erdős–Hajnal property).

[L2]

A graph is $\overline{K_{t}}$ -free exactly when its complement is $K_{t}$ -free, and the Erdős–Hajnal property is preserved by taking complements of graph classes ( $G$ is $H$ -free if and only if $\overline{G}$ is $\overline{H}$ -free, The complement of a graph class, A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants).

[L3]

If two graphs have the Erdős–Hajnal property, then substituting one for a vertex of the other preserves that property (Alon–Pach–Solymosi: if $H_{1}$ and $H_{2}$ have the Erdős–Hajnal property, so does the graph obtained from $H_{1}$ by substituting $H_{2}$ for a vertex, Substituting one graph for a vertex of another).

Proof

technique · direct

1.1

The complete graph $K_{t}$ has the Erdős–Hajnal property, since every $K_{t}$ -free class does by [L1].

2.1

The edgeless graph $\overline{K_{t}}$ has the Erdős–Hajnal property: by [L2] the class of $\overline{K_{t}}$ -free graphs is the complementary class of the $K_{t}$ -free graphs, so it has the same property.

step 1.1L2

3.1

Applying [L3] to $H$ and $K_{t}$ shows that substituting $K_{t}$ for $v$ preserves the Erdős–Hajnal property, and applying [L3] to $H$ and $\overline{K_{t}}$ shows the same for the edgeless graph.

step 1.1step 2.1L3∎

Depends on

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