AI 中文总结
本文将平面图强奇数色数的上界从388改进至368，还构造了强奇数色数为20的平面图，回应了相关研究问题。
AI 中文摘要
图的真着色被称为强奇着色，当且仅对于每个顶点\textit{v}及其开邻域\textit{N(v)}中出现的每种颜色，该颜色在\textit{N(v)}中出现奇数次；对应的最小颜色数即为强奇数色数，记为\textit{χₛₒ(G)}。Caro等人证明每个平面图的强奇数色数至多为388，Manattu等人后续构造了强奇数色数为17的平面图，并提出两个问题：是否存在更大的强奇数色数平面图，以及上界388能否改进。本文解决这些问题：首先，通过将无环2-连通平面多重图的辅助真面奇着色的颜色数上界从97改进至92，结合Caro等人的归约结果与四色定理，将一般平面图的强奇数色数上界改进为368；其次，给出一个强奇数色数为20的不同显式平面图构造，并提供该精确值的自包含证明。需说明的是，Goetze等人已于2025年5月发布的arXiv预印本中包含强奇数色数为20的平面图实例，故本文的构造并非对20该值的优先权主张，而是一个独立且经完全验证的构造，其值超过了Manattu等人提出问题所基于的17。
英文摘要
A proper coloring of a graph is called a strong odd coloring if, for every vertex \(v\) and every color appearing in the open neighborhood of \(v\), that color appears an odd number of times in \(N(v)\). The corresponding minimum number of colors is the strong odd chromatic number, denoted by \(χ_{\mathrm{so}}(G)\). Caro et al.~\cite{CaroPetrusevskiSkrekovskiTuzaStrongOdd} proved that every planar graph has strong odd chromatic number at most \(388\). Manattu et al.~\cite{ManattuVinayLakshmanan2026} later constructed planar graphs with strong odd chromatic number \(17\) and asked whether larger values are possible and whether the upper bound \(388\) can be improved. We address these questions as follows. First, we improve the general upper bound by proving that every planar graph \(G\) satisfies \(χ_{\mathrm{so}}(G)\le 368\). This follows by improving the auxiliary proper facially odd coloring bound for loopless \(2\)-connected plane multigraphs from \(97\) colors to \(92\) colors and combining this with the reduction of Caro et al. and the Four Color Theorem. Second, we give a different explicit planar construction with \(χ_{\mathrm{so}}(G)=20\), together with a self-contained proof of the exact value. We emphasize that Goetze et al.~\cite{GoetzeKluteKnauerParadaPenaUeckerdt2025} had already posted an arXiv preprint in May 2025 containing a planar example with strong odd chromatic number \(20\). Thus our construction is not a priority claim for the value \(20\), but rather an independent and fully verified construction whose value exceeds \(17\), the value that motivated Problem~1 of Manattu et al.
