ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Deleting one edge from $K_{3, 4}$ gives a triangle-free graph one edge below the Mantel threshold

Example

Delete any one edge from $K_{3, 4}$ . The resulting seven-vertex graph is triangle-free and has $11$ edges, exactly one fewer than $ex (7, K_{3})$ .

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

The complete bipartite graph $K_{A, B}$ has exactly all edges joining a vertex of $A$ to a vertex of $B$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

For every $n \in N$ , Mantel's theorem gives $ex (n, K_{3}) = ⌊ n^{2} / 4 ⌋$ , and a triangle-free $n$ -vertex graph attains equality exactly when it is the balanced complete bipartite graph up to isomorphism (Mantel's theorem: $ex (n, K_{3}) = ⌊ n^{2} / 4 ⌋$ , uniquely attained by $T_{n, 2}$ ).

Verification

technique · deletion preserves the forbidden-subgraph condition

1.1

Every edge of $K_{3, 4}$ crosses its two parts, so it has no triangle, and its edge count is $3 \cdot 4 = 12$ . Deleting an edge cannot create a triangle and changes the edge count to $11$ .

givenF1

2.1

Mantel's theorem gives threshold $12$ and identifies the unique equality graph as $K_{3, 4}$ up to isomorphism. The edge-deleted graph therefore lies exactly one edge below the equality case.

step 1.1givenF2∎

Depends on

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Deleting one edge from K3,4K3,4​ gives a triangle-free graph one edge below the Mantel threshold