Geodesic
Clusters in a multigraph with elevated density
The electronic journal of combinatorics, Tome 14 (2007)
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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.
DOI : 10.37236/4847
Classification : 05C15, 05C35 
@article{10_37236_4847,
     author = {Mark K. Goldberg},
     title = {Clusters in a multigraph with elevated density},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/4847},
     zbl = {1110.05032},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/4847/}
}
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%J The electronic journal of combinatorics
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Mark K. Goldberg. Clusters in a multigraph with elevated density. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/4847

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