Abstract
Liu and Xu (1998) and Ellingham, Nam and Voss (2002) independently showed that every k-edge-connected simple graph G has a spanning tree T such that for each vertex v, dT(v)≤⌈d(v)k⌉+2. In this paper we show that every k-edge-connected graph G has a spanning tree T such that for each vertex v, dT(v)≤⌈d(v)−2k⌉+2; also if G has k edge-disjoint spanning trees, then T can be found such that for each vertex v, dT(v)≤⌈d(v)−1k⌉+1. This result implies that every (r−1)-edge-connected r-regular graph (with r≥4) has a spanning Eulerian subgraph whose degrees lie in the set {2,4,6}; also reduces the edge-connectivity needed for some theorems due to Barát and Gerbner (2014) and Thomassen (2008, 2013). Moreover these bounds for finding spanning trees are sharp.
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