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Matthias Kriesell, Jens M. Schmidt

TL;DR
This paper investigates the existence and quantity of $k$-contractible edges in various types of $k$-connected graphs and their spanning trees, extending previous results and exploring optimality.
Contribution
It generalizes earlier findings on 3-contractible edges to broader classes of $k$-connected graphs and their spanning trees, providing new bounds and conditions.
Findings
Every spanning tree of a $k$-connected triangle-free graph has two $k$-contractible edges.
Spanning trees of $k$-connected graphs with high minimum degree have two $k$-contractible edges.
Certain DFS trees in $k$-connected graphs also contain two $k$-contractible edges.
Abstract
An edge in a -connected graph is called {\em -contractible} if the graph obtained from by contracting is -connected. Generalizing earlier results on -contractible edges in spanning trees of -connected graphs, we prove that (except for the graphs if ) (a) every spanning tree of a -connected triangle free graph has two -contractible edges, (b) every spanning tree of a -connected graph of minimum degree at least has two -contractible edges, (c) for , every DFS tree of a -connected graph of minimum degree at least has two -contractible edges, (d) every spanning tree of a cubic -connected graph nonisomorphic to has at least many -contractible edges, and (e) every DFS tree of a -connected graph nonisomorphic to , the prism, or…
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Taxonomy
TopicsGraph theory and applications
