New graceful diameter-6 trees by transfers
Matt Superdock

TL;DR
This paper proves that certain classes of diameter-6 trees, especially those with specific structural properties and odd number of children at internal vertices, are graceful, advancing understanding of the Graceful Tree Conjecture.
Contribution
It introduces new classes of diameter-6 trees that are proven to be graceful under specific structural conditions, extending previous results.
Findings
Diameter-6 complete trees with certain properties are graceful.
All depth-3 trees with internal vertices having odd children are graceful.
Conditions under which diameter-6 trees are proven to be graceful.
Abstract
Given a graph , a labeling of is an injective function . Under the labeling , the label of a vertex is , and the induced label of an edge is . The labeling is graceful if the labels of the vertices are , and the induced labels of the edges are distinct. The graph is graceful if it has a graceful labeling. The Graceful Tree Conjecture, introduced by Kotzig in the late 1960's, states that all trees are graceful. It is an open problem whether every diameter-6 tree has a graceful labeling. In this paper, we prove that if is a tree with central vertex and root , such that each vertex not in the last two levels has an odd number of children, and satisfies one of the following conditions (a)-(e), then has a graceful labeling with : (a) is a…
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Advanced Graph Theory Research · graph theory and CDMA systems
