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arXiv · 2608.21585

The Asayama-Matsumoto conjecture and a refined discrepancy bound

Alessio Basti·Tommaso Cremaschi
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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 $n\ge6$ and $n\not\equiv5\pmod6$, we prove the sharper bound $n-2\ceil{(n+2)/3}$. An exhaustive census of all $9{,}150$ non-isomorphic sphere triangulations with $4\le n\le12$ verifies the bounds and also confirms the sharper estimate at $n=11$, the first order in the remaining open residue class.

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Alessio Basti, Tommaso Cremaschi. 2026-08-21. The Asayama-Matsumoto conjecture and a refined discrepancy bound. https://arxiv.org/abs/2608.21585

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