Improved Bounds for Relaxed Graceful Trees

Abstract

We introduce left and right-layered trees as trees with a specific representation and define the excess of a tree. Applying these ideas, we show a range-relaxed graceful labeling which improves on the upper bound for maximum vertex label given by Van Bussel. For the case when the tree is a lobster of size m and diameter d, the labeling produces vertex labels no greater than $\frac{3}{2}m-\frac{1}{2}d$. Furthermore, we show that any lobster T with m edges and diameter d has an edge-relaxed graceful bipartite labeling with at least $\max \{\frac{3m-d+6}{4},\frac{5m+d+15}{8}\}$ of the edge weights distinct, which is an improvement on a bound given by Rosa and Širáň on the $\alpha$-size of trees, for $d<\frac{m+22}{7}$ and $d>\frac{5m-65}{7}$. We also show that there exists an edge-relaxed graceful labeling (not necessarily bipartite) with at least $\max \left\{ \frac{3}{4}m+\frac{d-\nu }{8}+\frac{3}{2},\nu \right\}$ of the edge weights distinct, where $\nu$ is twice the size of a partial matching of T. This is an improvement on the best known gracesize bound, for certain values of $\nu$ and d. We view these results as a step towards Bermond’s conjecture that all lobsters are graceful.