Dvořák–Norin–Rahman density conjecture for a K7-minus-two-edges minor

The conjecture asserts that sufficiently dense $5$-connected graphs contain the minor obtained from $K_7$ by deleting two independent edges; the sharp proposed threshold is $4n-9$ edges for $n\ge 7$.

Known results

• A weaker theorem gives the minor at the threshold $4n-7$ for $n\ge 6$.
• A rooted-minor theorem gives a rooted $K_5$ minor at $4n-10$ edges.
• The weaker density theorem implies that every graph excluding this minor is $6$-colorable.
• The sharp $4n-9$ statement was previously presented as a conjecture.

September 22, 2026 claimed proof

Caibing Chang, Zijian Deng, Qinfei Tang, and Caihong Yang report that their preprint proves the sharp density conjecture, together with a stronger rooted-minor statement and the extremal obstruction. This is an unrefereed preprint claim, and no independent verification or correction was found.

Current status (as of September 2026): The $4n-7$ theorem and related coloring result are established in the retrieved record, while the sharp $4n-9$ conjecture is claimed solved by an unrefereed preprint but remains unverified.