Erdős Problem #167

If $G$ is a graph with at most $k$ edge disjoint triangles then can $G$ be made triangle-free after removing at most $2k$ edges?

#167: [Er88]

graph theory

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A problem of Tuza. It is trivial that $G$ can be made triangle-free after removing at most $3k$ edges. The examples of $K_4$ and $K_5$ show that $2k$ would be best possible.

The trivial bound of $\leq 3k$ was improved to $\leq (3-\frac{3}{23}+o(1))k$ by Haxell [Ha99].

Kahn and Park [KaPa22] have proved this is true for random graphs.

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This page was last edited 13 October 2025.

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Additional thanks to: msellke

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