Rainbow spanning trees in abelian groups

Let ( A , + ) be a finite Abelian group. Take the elements of A to be vertices of a complete graph and color the edge ab with a + b . A tree in A is rainbow colored provided all of its edges have different colors. In this paper, we study conditions that regulate whether or not a given tree can be realized as a rainbow spanning subtree of an Abelian group of the same order. For example, let C [ h 1 , ⋯ , h s ] denote the caterpillar with s spine vertices and with h i hairs on the i th spine vertex. We characterize, by means of divisibility conditions, when a caterpillar of type C [ k , ℓ ] , C [ k , 0 , ℓ ] or of type C [ k , 0 , 0 , ℓ ] embeds as a rainbow spanning tree in a group of the same order. We also show that embeddability as a rainbow spanning tree is not a local condition. That is, given any tree T and sufficiently large non-cyclic group A , some trees of order A that contain T as a subtree do embed as rainbow spanning trees in A , and some do not. For non-Boolean groups A of order at most 20, we give a complete catalogue of all trees that fail to embed as rainbow spanning trees of A . We also show that all rainbow spanning trees in A can be obtained from the star with center 0 through a simple pivoting procedure.