On covering vertices of a graph by trees. (English summary) Discrete Math. 308 (2008), no. 19, 4414–4418. This paper deals with the following problem: Let G be a graph, and k ≥ 1. Determine the minimum number s of trees T1,..., Ts, δ(Ti) ≤ k, i = 1,..., s, covering all vertices of G. The authors conjecture: Let G be a connected graph, and k ≥ 2. Then the vertices of G can be covered by s ≤ n−δ ⌈δ(k−1)+1 ⌉ edge-disjoint trees of maximum degree ≤ k. As a support for the conjecture, they prove the statement in the cases where δ = 1 and all k ≥ 2, and where k = 2 and δ ≥ 2. The authors also conjecture the following statement of covering vertices of a graph by paths: n d+3+o(1) ⌉ Let G be a connected d-regular graph. Then the vertices of G can be covered by ⌈ vertex-disjoint paths. If these conjectures are true, they provide the best possible bound on the numbers of edge-disjoint trees and of vertex-disjoint paths. Reviewed by Kenji Kimur
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