An Infinite Series of Surfaces with Known 1-Chromatic Number

The 1-chromatic numberχ1(S) of a surfaceSis 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⩾0, thenχ1(N8(2n+1)2)=R(N8(2n+1)2) whereR(S)=⌊12 (9+81−32E(S))⌋ is Ringelʹs upper bound forχ1(S),E(S) is the Euler characteristic ofS, andN8(2n+1)2is the nonorientable surface of genus 8(2n+1)2. By Dirichletʹs theorem the arithmetic progression 4n+3,n=1, 2, 3, …, contains an infinite number of prime integers. As a result the first known infinite series of surfaces with known 1-chromatic number is obtained.