A short solution of Heawood's empire problem on the torus and implications for higher genus (Q1970588)

\textit{P. J. Heawood} [Q. J. Pure Appl. Math. 24, 332-338 (1890)] establishing an upper bound that, when specialized to the torus, shows that \(6c+1\) colors suffice to color any toroidal map for which each country has at most \(c\) components. \textit{B. Jackson} and \textit{G. Ringel} [J. Reine Angew. Math. 347, 146-153 (1984; Zbl 0519.05028)] showed that \(6c+1\) colors are also necessary. The present author provides a short proof of the necessity, using elementary constructions to produce a \(c\)-uniform complete toroidal multimap with \(6c+ 1\) empires, for every \(c\geq 1\).