On covering vertices of a graph by trees

Abstract

The purpose of this paper is to initiate study of the following problem: Let G be a graph, and k ⩾ 1 . Determine the minimum number s of trees T 1 , … , T s , Δ ( T i ) ⩽ k , i = 1 , … , s , covering all vertices of G . We 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 we prove the statement for some values of δ and k .