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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 $k$-connected graph $G$ is called {\em $k$-contractible} if the graph $G/e$ obtained from $G$ by contracting $e$ is $k$-connected. Generalizing earlier results on $3$-contractible edges in spanning trees of $3$-connected graphs, we prove that (except for the graphs $K_{k+1}$ if $k \in \{1,2\}$) (a) every spanning tree of a $k$-connected triangle free graph has two $k$-contractible edges, (b) every spanning tree of a $k$-connected graph of minimum degree at least $\frac{3}{2}k-1$ has two $k$-contractible edges, (c) for $k>3$, every DFS tree of a $k$-connected graph of minimum degree at least $\frac{3}{2}k-\frac{3}{2}$ has two $k$-contractible edges, (d) every spanning tree of a cubic $3$-connected graph nonisomorphic to $K_4$ has at least $\frac{1}{3}|V(G)|-1$ many $3$-contractible edges, and (e) every DFS tree of a $3$-connected graph nonisomorphic to $K_4$, the prism, or the prism plus a single edge has two 3-contractible edges. We also discuss in which sense these theorems are best possible.