A Generalization of the Graph Packing Theorems of Sauer-Spencer and Brandt

TL;DR

This paper generalizes key graph packing theorems, providing a unified framework and characterizing extremal cases for packing a forest into a graph based on degree and leaf parameters.

Contribution

It introduces a unified theorem that generalizes Sauer-Spencer and Brandt's results, and characterizes extremal graphs for forest packing.

Findings

Unified graph packing theorem encompassing Sauer-Spencer and Brandt's results.

Characterization of extremal graphs when packing a forest into a graph.

Conditions under which packing is possible or fails.

Abstract

We prove a common generalization of the celebrated Sauer-Spencer packing theorem and a theorem of Brandt concerning finding a copy of a tree inside a graph. This proof leads to the characterization of the extremal graphs in the case of Brandt's theorem: If $G$ is a graph and $F$ is a forest, both on $n$ vertices, and $3\Delta(G)+\ell^*(F)\leq n$, then $G$ and $F$ pack unless $n$ is even, $G=\frac{n}{2}K_2$ and $F=K_{1,n-1}$; where $\ell^*(F)$ is the difference between the number of leaves and twice the number of nontrivial components of $F$.