An infinite series of surfaces with known 1-chromatic number
The 1-chromatic number \(\chi_1(S)\) of a surface \(S\) is the maximum chromatic number of all graphs which can be drawn on the surface so that each edge is crossed by no more than one other edge. It is proved that if \(4n+3\) is a prime number \((n\geq 0)\) then \(\chi_1(N_{8(2n+1)^2})= R(N_{8(2n+1)^2})\), where \(R(S)=\lfloor \frac12 (9+\sqrt {81-32E(S))}\rfloor\) is Ringel's upper bound for \(\chi_1(S)\), \(E(S)\) is the Euler characteristic of \(S\) and \(N_{8(2n+1)^2}\) is the nonorientable surface of genus \(8(2n+1)^2\). By Dirichlet's theorem the arithmetic progression \(4n+3\) \((n=1,2,3,\dots)\) contains an infinite number of prime integers. As a result the first known infinite series of surfaces with known 1-chromatic number is obtained.
scientific article; zbMATH DE number 1060753