A subdivided 6-clique breaks a conjectured sum bound for independent monophonic-position colorings

A set of vertices of a graph is in monophonic position if no induced path contains three of its vertices. Ullas Chandran S. V., Gabriele Di Stefano, S. Haritha, Elias John Thomas and James Tuite introduced colorings whose color classes are such sets, and the variant whose classes are also independent. They showed that the monophonic-position chromatic numbers of an $n$-vertex graph and of its complement sum to at most $n+1$, and conjectured the same Nordhaus–Gaddum bound for the independent variant; they report a computer search by Grahame Erskine confirming their conjecture for all graphs with at most ten vertices. We show that the independent monophonic sum bound fails. Subdividing every edge of the complete graph $K_6$ once gives a graph $H$ on 21 vertices that needs at least eight independent monophonic-position color classes, while its complement needs at least fifteen, so the sum is at least 23. The proof classifies the independent monophonic-position sets of $H$ and shows by counting that seven color classes cannot cover $H$. The lower bounds are also confirmed by SAT-solver proofs checked with a formally verified checker. This work has not yet been reviewed by experts. The proofs, the code and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.