Decomposition of Complete Graphs into Arbitrary Trees

Abstract

In this paper, we propose a new conjecture that the complete graph $$K_{4m+1}$$ can be decomposed into copies of two arbitrary trees, each of size $$m, m \ge 1$$ . To support this conjecture we prove that the complete graph $$K_{4cm+1}$$ can be decomposed into copies of an arbitrary tree with m edges and copies of the graph H, where H is either a path with m edges or a star with m edges and where c is any positive integer. Further, we discuss related open problems.