The Asayama-Matsumoto conjecture and a refined discrepancy bound

Abstract

Every $n$-vertex plane triangulation admits a polychromatic red--blue vertex coloring with discrepancy at most $n-2\ceil{n/3}\le\floor{n/3}$, resolving a conjecture of Asayama and Matsumoto. The proof follows by combining Loyola et al. 2026 and Kawarabayashi et al. 2026. For $n\ge 6$ and $n\not\equiv 5(\mathrm{mod}6)$, we prove the sharper bound $n-2\ceil{(n+2)/3}$. An exhaustive census of all $9,150$ non-isomorphic sphere triangulations with $4\le n\le 12$ verifies the bounds and also confirms the sharper estimate at $n=11$, the first order in the remaining open residue class.