Trees having an even or quasi even degree sequence are graceful

Abstract

A rooted tree with diameter $D$ is said to have an even degree sequence if every vertex has even degree except for one root and the leaves, which are in the last level $\lfloor D/2\rfloor$. The degree sequence is said to be quasi even if every vertex has even degree except for one root, every vertex in level $\lfloor D/2\rfloor -1$ and the leaves, which are in the last level $\lfloor D/2\rfloor$. Hrnčiar and Haviar [P. Hrnčiar, A. Haviar, All trees of diameter five are graceful, Discrete Math. 233 (2001) 133–150] give a method to construct a graceful labeling for every tree with diameter five. Based upon their method we prove that every tree having an even or quasi even degree sequence is graceful. To do that we find for a tree of even diameter and rooted in its central vertex $t$ of degree $\delta \left(t\right)$ up to $\delta \left(t\right)!$ graceful labelings if the tree has an even or quasi even degree sequence.