Labeled Packing of Non Star Tree into its Fifth Power and Sixth Power

TL;DR

This paper establishes new bounds for labeled packings of non star trees into their fifth and sixth powers, improving understanding of graph packing with specific label constraints.

Contribution

It introduces novel bounds for labeled packings of non star trees into their fifth and sixth powers, extending previous graph packing results.

Findings

Labeled packing of non star trees into T^6 with m_T + ceil((n - m_T)/5) labels.

Labeled packing of non star trees into T^5 with m_T + 1 labels.

Labeled packing of paths into their fourth power with ceil(n/4) labels.

Abstract

In this paper we prove that we can find a labeled packing of a non star tree $T$ into $T^6$ with $m_T+\lceil\frac{n-m_T}{5}\rceil$ labels, where $n$ is the number of vertices of $T$ and $m_T$ is the maximum number of leaves that can be removed from $T$ in such a way that the obtained graph is a non star tree. Also, we prove that we can find a labeled packing of a non star tree $T$ into $T^5$ with $m_T+1$ labels and a labeled packing of a path $P_n$, $n\geq 4$, into $P_n^4$ with $\lceil \frac{n}{4}\rceil$ labels.