# Toughness and spanning trees in $K_4$-minor-free graphs

## TL;DR

This paper proves that connected $K_4$-minor-free graphs with certain toughness conditions contain spanning trees with bounded degrees, extending known results for general graphs by leveraging the structure of these specific graph classes.

## Contribution

It establishes new sufficient conditions for the existence of spanning $f$-trees in $K_4$-minor-free graphs, strengthening previous results for broader classes of graphs.

## Key findings

- Connected $K_4$-minor-free graphs have spanning $f$-trees under specific component conditions.
- Toughness conditions imply the existence of spanning $k$-trees in these graphs.
- The results do not extend to all planar graphs, highlighting the importance of the $K_4$-minor-free restriction.

## Abstract

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the number of components of a graph $G$. We show that if $G$ is a connected $K_4$-minor-free graph and $$ c(G-S) \;\le\; \sum_{v \in S} (f(v)-1) \quad\hbox{for all $S \subseteq V(G)$ with $S \ne \emptyset$} $$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a $\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree. These results are stronger than results for general graphs due to Win (for $k$-trees) and Ellingham, Nam and Voss (for $f$-trees). The $K_4$-minor-free graphs form a subclass of planar graphs, and are identical to graphs of treewidth at most $2$, and also to graphs whose blocks are series-parallel. We provide examples to show that the inequality above cannot be relaxed by adding $1$ to the right-hand side, and also to show that our result does not hold for general planar graphs. Our proof uses a technique where we incorporate toughness-related information into weights associated with vertices and cutsets.