Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II

Morteza Hasanvand

Published 2017-02-20Version 1

Let $G$ be a connected graph with $X\subseteq V(G)$ and with the spanning forest $F$. Let $\lambda\in [0,1]$ be a real number and let $\eta:X\rightarrow (\lambda,\infty)$ be a real function. In this paper, we show that if for all $S\subseteq X$, $\omega(G\setminus S)< 1+\sum_{v\in S}\big(\eta(v)-2\big)+2\,-\lambda(|E_S(G)|+1)$, then $G$ has a spanning tree $T$ containing $F$ such that for each vertex $v\in X$, $d_T(v)\le \lceil\eta(v)-\lambda\rceil+\max\{0,d_F(v)-1\}$, where $\omega(G\setminus S)$ denotes the number of components of $G\setminus S$ and $E_S(G)$ denotes the set of edges of $G$ whose ends lie in the set $S$. This is an improvement of several results and the condition is best possible. Next, we show that every bipartite graph $G$ with $k$ edge-disjoint spanning trees and with the bipartition $(A,B)$ has a spanning tree $T$ such that for each vertex $v\in A$, $d_T(v)\le \lceil \frac{d_G(v)}{k}\rceil$. This reduces the required edge-connectivity of a result due to Bar\'at and Gerbner (2014) toward decomposing a graph into isomorphic copies of a fixed tree. Finally, we show that every $4k$-edge-connected graph with the upper bound of $4k+6$ on its degrees has a spanning Eulerian subgraph with the upper bound of $6$ on its degrees.