In this paper, we shall show that every quadrangulation on a nonorientable closed surface with sufficiently large representativity has chromatic number 2, 3 or 4 and characterize those for each value, discussing an algebraic invariant called a cycle parity. In particular, we shall prove that such a quadrangulation is 4-chromatic if and only if it has an odd cycle which cuts open the host surface into an orientable surface.
- Chromatic number
- Cycle parity
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics