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 (Γ−3)/(Γ−∆−1). Finally, we show that two well-known lower bounds for the chromatic index of a multigraph are equal.

• Type: Journal
• Year: 2007
• Where: COMBINATORICS
• Authors: Mark K. Goldberg
• Keywords: COMBINATORICS 2007, Dense Sets, Maximum Vertex Degree, Minimal Clusters