Every cohomology class of M_{g,m} (for 2g+m≥3, m≥1) is represented by a ribbon quiver with at most four-valent vertices, and this bound is sharp.
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A low-valence ribbon graph complex computing the cohomology of $M_{g,m}$
Every cohomology class of M_{g,m} (for 2g+m≥3, m≥1) is represented by a ribbon quiver with at most four-valent vertices, and this bound is sharp.