The empire problem in even embeddings on closed surfaces with

Let M be a map on a closed surface F 2 and suppose that each country of the map has at most r disjoint connected regions. Such a map is called an r -pire map on F 2 . In 1890, Heawood proved that the countries of M can be properly colored with ⌊ ( 6 r + 1 + ( 6 r + 1 ) 2 − 24 ε ) / 2 ⌋ colors, where ε is the Euler characteristic of F 2 . Also, he conjectured that this is best possible except for the case ( ε , r ) = ( 2 , 1 ) , and now it is proved for all cases where ε ≥ 0 and some cases where ε < 0 . l a graph on F 2 an even embedding if it has no faces of boundary length odd. In this paper, we consider the r -pire maps whose underlying graphs are even embeddings on F 2 . In my recent paper, it was proved that such a map can be properly colored with ⌊ ( 4 r + 1 + ( 4 r + 1 ) 2 − 16 ε ) / 2 ⌋ colors. In this paper, we show the best possibility of this value for some cases where ε ≤ 0 .