A possibly infinite series of surfaces with known 1-chromatic number Original Research Article

Let χ1(S) be the maximum chromatic number for all graphs which can be drawn on a surface S so that each edge is crossed by no more than one other edge. It is proved that if 2 is a primitive root modulo 4n + 5, n ⩾ 1, n ≢ 1 mod 3, then χ1(N8n2), where F(S) = ⦜12(9 + √81 − 32E(S))⊥ is Ringelʹs upper bound for χ1(S), E(S) is the Euler characteristic of S and N8n2 is the nonorientable surface of genus 8n2. Some number-theoretic arguments are advanced in favour of that it may be an infinite number of such integers n that 2 is a primitive root modulo 4n + 5, n ⩾ 1, n ≢ 1 mod 3.