Spanning trees and spanning Eulerian subgraphs with small degrees

Packing of 2-Connected Spanning Subgraphs and Spanning Trees

We prove that every (6k + 2`, 2k)-connected simple graph contains k rigid and ` connected edgedisjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach p...

Packing of rigid spanning subgraphs and spanning trees

We prove that every (6k + 2`, 2k)-connected simple graph contains k rigid and ` connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach ...

Spanning trees with minimum weighted degrees

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this prob...

Spanning Trees with Vertices Having Large Degrees

Let G be a connected simple graph, and let f be a mapping from V (G) to the set of integers. This paper is concerned with the existence of a spanning tree in which each vertex v has degree at least f(v). We show that if ∣∣ΓG(S)∣∣− f(S)+ |S| ≥ 1 for any nonempty subset S ⊆ L, then a connected graph G has a spanning tree such that dT (x) ≥ f(x) for all x ∈ V (G), where ΓG(S) is the set of neighbo...

Spanning trees in subgraphs of lattices