Vertices of degree k in edge-minimal, k-edge-connected graphs
Carl Kingsford, Guillaume Mar\c{c}ais
TL;DR
This paper extends Halin's theorem to edge-minimal, k-edge-connected graphs, proving such graphs always contain two vertices of degree k, thereby advancing understanding of their structural properties.
Contribution
It proves that in edge-minimal, k-edge-connected graphs, there are always two vertices of degree k, generalizing Halin's theorem from vertex connectivity to edge connectivity.
Findings
Edge-minimal, k-edge-connected graphs have at least two vertices of degree k.
The result generalizes Halin's theorem from vertex to edge connectivity.
Provides new insights into the structure of minimally edge-connected graphs.
Abstract
Halin showed that every edge minimal, k-vertex connected graph has a vertex of degree k. In this note, we prove the analogue to Halin's theorem for edge-minimal, k-edge-connected graphs. We show there are two vertices of degree k in every edge-minimal, k-edge-connected graph.
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Taxonomy
TopicsAdvanced Graph Theory Research · Interconnection Networks and Systems · Graph theory and applications