Abstract
Let χ 1( 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 4 n + 5, n ⩾ 1, n ≢ 1 mod 3, then χ 1( N 8 n 2 ), where F(S) = ⦜ 1 2 (9 + √81 − 32E(S))⊥ is Ringel's upper bound for χ 1( S), E( S) is the Euler characteristic of S and N 8 n 2 is the nonorientable surface of genus 8 n 2. 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 4 n + 5, n ⩾ 1, n ≢ 1 mod 3.
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