Some methods for constructing some classes of graceful uniform trees

Abstract

<p>A tree <span class="math"><em>T</em>(<em>V</em>, <em>E</em>)</span> is <span><em>graceful</em></span> if there exists an injective function <span class="math"><em>f</em></span> from the vertex set <span class="math"><em>V</em>(<em>T</em>)</span> into the set <span class="math">{0, 1, 2, ..., ∣<em>V</em>∣ − 1}</span> which induces a bijective function <span class="math"><em>f</em>ʹ</span> from the edge set <span class="math"><em>E</em>(<em>T</em>)</span> onto the set <span class="math">{1, 2, ..., ∣<em>E</em>∣}</span>, with <span class="math"><em>f</em>ʹ(<em>u</em><em>v</em>) = ∣<em>f</em>(<em>u</em>) − <em>f</em>(<em>v</em>)∣</span> for every edge <span class="math">{<em>u</em>, <em>v</em>} ∈ <em>E</em></span>. Motivated by the conjecture of Alexander Rosa (see) saying that all trees are graceful, a lot of works have addressed gracefulness of some trees. In this paper we show that some uniform trees are graceful. This results will extend the list of graceful trees.</p>