Clusters in a Multigraph with Elevated Density

In this paper, we prove that in a multigraph whose density F exceeds the maximum vertex degree A, 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.