LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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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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A rooted stable-tooth comb with a cross-edge between two blocks contains an induced five-cycle

Statement

Let $G$ be a finite graph containing a rooted stable-tooth comb

$(v, ((a_{i}, B_{i}) : 1 \leq i \leq t)) .$

If $1 \leq i < j \leq t$ and there are vertices $b_{i} \in B_{i}$ and $b_{j} \in B_{j}$ with $b_{i} b_{j} \in E (G)$ , then the induced subgraph on ${v, a_{i}, b_{i}, b_{j}, a_{j}}$ is isomorphic to $C_{5}$ .

Facts & Assumptions

Given: A rooted stable-tooth comb $(v, ((a_{i}, B_{i}) : 1 \leq i \leq t))$ in a finite graph $G$ , indices $i < j$ , and adjacent vertices $b_{i} \in B_{i}$ , $b_{j} \in B_{j}$ .

[L1]

In a rooted stable-tooth comb, each tooth is adjacent to every vertex of its own block, anticomplete to every other block, the teeth form a stable set, and the root is adjacent to all teeth and anticomplete to every block (A rooted stable-tooth comb).

[L2]

An induced copy of $C_{5}$ is a five-vertex set whose induced subgraph is isomorphic to the cycle graph on five vertices (Induced embeddings and induced copies of a graph, Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Proof

technique · direct

1.1

By [L1], the edges $v a_{i}$ , $v a_{j}$ , $a_{i} b_{i}$ , $a_{j} b_{j}$ , and $b_{i} b_{j}$ are present. The same definition excludes every other edge among ${v, a_{i}, b_{i}, b_{j}, a_{j}}$ : the teeth $a_{i}, a_{j}$ are nonadjacent, the root $v$ is anticomplete to the blocks, and each tooth is anticomplete to the other tooth's block.

2.1

Therefore the cyclic order $v - a_{i} - b_{i} - b_{j} - a_{j} - v$ uses exactly the edges of the induced subgraph on ${v, a_{i}, b_{i}, b_{j}, a_{j}}$ . By [L2], that induced subgraph is a copy of $C_{5}$ .

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

12 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, proof of Theorem 4.4 (standard reference, not scraped)