0-rotatability of classes of rooted symmetric trees. Are rooted symmetric trees 0-rotatable?

Abstract

A graceful labelling of a tree T = (V,E), where V is the set of vertices of the tree and E is its edge set, is a bijective function f from V to the set consisting of the numbers 0, 1, ... |E| inclusive, such that if edge uv is assigned the value |f(u)-f(v)| then the edge labels are distinct numbers of the set consisting of the numbers 1, 2, ..., |E| inclusive. A tree is said to be 0-roratable if for any of its vertices there is a graceful labelling that assigns the label 0 to that vertex. A rooted symmetric tree is a tree in which all vertices at the same level from root vertex have the same degree. It was known since 1979 that rooted symmetric trees are graceful and an algebraic definition of graceful labelling of this class of trees was found by the author. In this paper we prove that rooted symmetric trees with at most 3 levels (including root vertex) are 0-rotatable. We also prove that symmetric spider trees with leg length at most 3 and symmetric banana trees, both of which are classes of rooted symmetric trees with 4 levels, are 0-rotatable. Based on these results, we conjecture that all spiders are 0-rotatable and raise the more general question whether all symmetric rooted trees are 0-rotatable.