The Asayama-Matsumoto conjecture and a refined discrepancy bound

Every $n$-vertex plane triangulation admits a polychromatic red–blue vertex coloring with discrepancy at most $n-2\text{ceil}n\mathrm{/}3\le \text{floor}n\mathrm{/}3$, 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 $n\equiv ̸5(\mathrm{m}\mathrm{o}\mathrm{d}6)$, we prove the sharper bound $n-2\text{ceil}(n+2)\mathrm{/}3$. An exhaustive census of all $\mathrm{9,150}$ 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.