Decompositions of graphs into trees

Let θ be a family of graphs. By a θ‐decomposition of a graph G we mean a partition λ of the edge set of G such that every F ϵ π spans in G a subgraph isomorphic to a graph in θ. In this paper we state the following conjecture: If T 1 and T 2 are two trees having relatively prime sizes then there exists c = c(T 1 T 2 ) such that every graph G satisfying the condition δ(G) ⩾ c has a { T 1 , T 2 }‐decom‐position. We prove this conjecture for some special pairs of trees. In particular, we prove it in the following cases: (i) T 1 and T 2 are stars having relatively prime sizes; (ii) T 1 and T 2 are paths having relatively prime sizes; and. (iii) T 1 = T 2 ‐ {v}, where v is a terminal vertex in T 2 .