Improved Bounds for Relaxed Graceful Trees

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 \v{S}ir\'{a}\v{n} 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 gracesize bound of Rosa and \v{S}ir\'{a}\v{n} for certain values of $\nu$ and $d$. We view these results as a step towards Bermond's conjecture.