Clusters in a Multigraph with Elevated Density

Mark K. Goldberg

Abstract


In this paper, we prove that in a multigraph whose density $\Gamma$ exceeds the maxiimum vertex degree $\Delta$, the collection of minimal clusters (maximally dense sets of vertices) is cycle-free. We also prove that for multigraphs with $\Gamma\gt\Delta+1$, 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.


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