An Improved Upper Bound for the Strong Odd Chromatic Number of Planar Graphs

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 $\chi_{\mathrm{so}}(G)$. 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 $\chi_{\mathrm{so}}(G)\le 368$. 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 $\chi_{\mathrm{so}}(G)=20$, 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.