A Toughness Condition for a Spanning Tree With Bounded Total Excesses

Abstract

Let $k$ be an integer with $k \ge 2$. In terms of the toughness of a graph, Win gave a sufficient condition for the existence of a spanning $k$-tree, that is, a spanning tree in which the maximum degree is at most $k$. For a spanning tree $T$ of a graph $G$, we define the total excess $\mathrm{te}(T,k)$ of $T$ from $k$ as $\mathrm{te}(T,k) := \sum _{v \in V(T)} \max \{d_T(v)-k, 0\}$, where $V(T)$ is the vertex set of $T$ and $d_T(v)$ is the degree of a vertex $v$ in $T$. Enomoto, Ohnishi and Ota extended Win’s result by giving a condition that guarantees the existence of a spanning tree $T$ with a bounded total excess. Here we further extend the result as follows. Let $\omega (G)$ be the number of components of a graph $G$. Let $k_1,k_2, \ldots ,k_p$ be $p$ integers with $k_1 \ge k_2 \ge \cdots \ge k_p \ge 2$, let $t_1,t_2, \ldots ,t_p$ be $p$ nonnegative integers, and let $G$ be a connected graph. If $G$ satisfies that $\omega (G-S) \le (k_i - 2)|S| + 2 +t_i$ for every $1 \le i \le p$ and every $S \subset V(G)$, then $G$ has a spanning tree $T$ such that $\mathrm{te}(T,k_i) \le t_i$ for every $1 \le i \le p$.