arXiv subjects

Rafael I. Rofa

Publications and source records attributed to Rafael I. Rofa.

3 recordsLinked to original sources

arXiv2023

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

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.

math.CO↗

arXiv2021

A Graceful Algebraic Function Labelling of Rooted Symmetric Trees

Let T=(V,E) be a tree with vertex set V and edge set E. A graceful labelling f of T is an injective function f from V into {0, 1, ..., |E|} such that if edge uv is assigned the label g(uv)=|f(u)-f(v)| then the function g from E into {1, ..., |E|} is also injective (that is all edge labels are distinct). A rooted symmetric tree is a tree in which all vertices at the same level from root vertex have the same degree. It has been known since 1979 that rooted symmetric trees are graceful. However, the proofs that have been presented for this fact are either indirect inductive proofs or algorithmic descriptive proofs showing the many separate steps involved in labelling the vertices. Given a rooted symmetric tree, we find a graceful labelling f of T in the form of a direct algebraic function that algebraically maps each vertex of T to a unique label. Interestingly, f turns out to be a generalisation of the way a path is canonically gracefully labelled as algorithmically described by A. Rosa who is a pioneer in this field of research. The algebraic function that defines the graceful labelling is used to show that a class of rooted symmetric trees that contains the class of binomial trees has weakly α-labelling and it can provide a concise practical, computational way of producing graceful labelling of large rooted symmetric trees in, relatively, minimal time.

math.GM↗

arXiv2017

On Number Representation

Place value numbers, such as the binary or decimal numbers can be represented by the end vertices (leaf or pendant vertices) of rooted symmetrical trees. Numbers that consist of at most a fixed number of digits are represented by vertices that are equidistant from the root vertex and the corresponding number representations do not depend on the distance from the root vertex. In this paper, we introduce number systems that are representable by all vertices of a rooted symmetric tree in such a way that the number representation of each vertex depends on its distance from the root vertex.

math.GM↗