In this paper, we prove that in a multigraph whose density $\Gamma$ exceeds the maximum vertex degree $\Delta$, the collection of minimal clusters (maximally dense sets of vertices) is cycle-free. We also prove that for multigraphs with $\Gamma > \Delta + 1$, the size of any cluster is bounded from the above by $(\Gamma - 3)/ (\Gamma - \Delta -1)$. Finally, we show that two well-known lower bounds for the chromatic index of a multigraph are equal.