Embedding a Forest in a Graph

TL;DR

This paper proves a condition under which any forest with p trees can be embedded into a graph based on the graph's size and minimum degree, extending understanding of graph embeddings.

Contribution

It establishes a new minimum degree and size condition for embedding forests with multiple trees into arbitrary graphs.

Findings

Embedding condition depends on total size and number of trees

Minimum degree threshold is tight for embedding forests

Applicable to graphs with arbitrary structures

Abstract

For \math{p\ge 1}, we prove that every forest with \math{p} trees whose sizes are $a_1,..., a_p$ can be embedded in any graph containing at least $\sum_{i=1}^p (a_i + 1)$ vertices and having a minimum degree at least $\sum_{i=1}^p a_i$.