A counterexample to a conjecture of Ghosh

TL;DR

The paper provides counterexamples to Ghosh's conjecture, demonstrating specific trees that do not admit certain graceful labelings, thus challenging previous assumptions in graph labeling theory.

Contribution

It introduces explicit counterexamples of lobster trees and trees with specific diameters that defy existing labeling conjectures, advancing understanding of graph labeling limitations.

Findings

Existence of a lobster tree of diameter less than 6 with no alpha-labeling

A tree of diameter 4 with an even degree central vertex lacking a maximum label in graceful labeling

Counterexamples challenge prior conjectures in graph labeling theory

Abstract

We answer two questions of Shamik Ghosh in the negative. We show that there exists a lobster tree of diameter less than 6 which accepts no alpha-labeling with two central vertices labeled by the critical number and the maximum vertex label. We also show a simple example of a tree of diameter 4, with an even degree central vertex which does not accept a maximum label in any graceful labeling.