Clusters in a Multigraph with Elevated Density

In this paper, we prove that in a multigraph whose density Γ exceeds the maximum vertex degree ∆, the collection of minimal clusters (maximally dense sets of vertices) is cycle-free. We also prove that for multigraphs with Γ > ∆ + 1, the size of any cluster is bounded from the above by a function, which depends on ∆ and Γ only. Finally, we show that two well-known lower bounds for the chromatic index of a multigraph are identical.