CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The graph $H_{0}$ has the Erdős-Hajnal property

Statement

The graph $H_{0}$ has the Erdős-Hajnal property.

Facts & Assumptions

Given: The graph $H_{0}$ .

[L1]

Every graph on at most three vertices has the Erdős-Hajnal property (Every graph on at most three vertices has the Erdős–Hajnal property).

[L2]

The graph $C_{5}$ has the Erdős-Hajnal property (The five-cycle has the Erdős-Hajnal property).

[L3]

Substitution of graphs is defined by replacing one vertex of a graph by a second graph and inheriting the original adjacency pattern (Substituting one graph for a vertex of another).

[L4]

Substitution preserves the Erdős-Hajnal 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).

[F1]

If $K_{2}$ has vertices $u, v$ and one substitutes $C_{5}$ for $v$ , then the new graph consists of the $C_{5}$ rim together with the remaining vertex $u$ adjacent to every rim vertex. This is exactly the five-wheel $H_{0}$ .

Proof

technique · direct

1.1

By [L1], the two-vertex graph $K_{2}$ has the Erdős-Hajnal property, and by [L2] so does $C_{5}$ .

L1L2

2.1

By [F1] and [L3], the graph $H_{0}$ is obtained by substituting $C_{5}$ for one vertex of $K_{2}$ . Therefore [L4] applies to the two graphs of step 1.1 and gives the Erdős-Hajnal property for $H_{0}$ .

step 1.1L3L4F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Figure 7 and Lemma 6.3 preface (standard reference, not scraped)

The graph H0H0​ has the Erdős-Hajnal property